Home / Math and Physics Files / Math / math misc

Ricci flow

PDF · 328 pages · 2.1 MB

A published long paper by Huai-Dong Cao and Xi-Ping Zhu (Asian J. Math. 10(2), June 2006), filed in Phil's math misc folder. It presents the Hamilton-Perelman theory of Ricci flow: evolution equations, maximum principles and Li-Yau-Hamilton estimates, Perelman's reduced volume and no local collapsing, singularity formation, ancient kappa-solutions, and Ricci flow with surgery on three-manifolds, ending with geometrization. No annotations by Phil are evident in the text shown.

AI-written summary; may contain errors.

Extracted text (machine-read; may contain errors)

ASIANJ. MATH. c/circleco√yrt2006 International Press Vol. 10, No. 2, pp. 165–492, June 2006 001 A COMPLETE PROOF OF THE POINCAR ´E AND GEOMETRIZATION CONJECTURES – APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOW∗ HUAI-DONG CAO†AND XI-PING ZHU‡ Abstract. In this paper, we give a complete proof of the Poincar´ e and th e geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crownin g achievement of the Hamilton-Perelman theory of Ricci ﬂow. Key words. Ricci ﬂow, Ricci ﬂow with surgery, Hamilton-Perelman theor y, Poincar´ e Conjec- ture, geometrization of 3-manifolds AMS subject classiﬁcations. 53C21, 53C44 CONTENTS Introduction 167 1 Evolution Equations 172 1.1 The Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 1.2 Short-time Existence and Uniqueness . . . . . . . . . . . . . . . . . . . 177 1.3 Evolution of Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . 1 83 1.4 Derivative Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 90 1.5 Variational Structure and Dynamic Property . . . . . . . . . . . . . . 199 2 Maximum Principle and Li-Yau-Hamilton Inequalities 210 2.1 Preserving Positive Curvature . . . . . . . . . . . . . . . . . . . . . . . 210 2.2 Strong Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . 21 3 2.3 Advanced Maximum Principle for Tensors . . . . . . . . . . . . . . . . 217 2.4 Hamilton-Ivey Curvature Pinching Estimate . . . . . . . . . . . . . . . 223 2.5 Li-Yau-Hamilton Estimates . . . . . . . . . . . . . . . . . . . . . . . . 226 2.6 Perelman’s Estimate for Conjugate Heat Equations . . . . . . . . . . . 234 3 Perelman’s Reduced Volume 239 3.1 Riemannian Formalism in Potentially Inﬁnite Dimension s . . . . . . . 239 3.2 Comparison Theorems for Perelman’s Reduced Volume . . . . . . . . . 243 3.3 No Local Collapsing Theorem I . . . . . . . . . . . . . . . . . . . . . . 2 55 3.4 No Local Collapsing Theorem II . . . . . . . . . . . . . . . . . . . . . 2 61 4 Formation of Singularities 267 4.1 Cheeger Type Compactness . . . . . . . . . . . . . . . . . . . . . . . . 26 7 4.2 Injectivity Radius Estimates . . . . . . . . . . . . . . . . . . . . . . . . 286 4.3 Limiting Singularity Models . . . . . . . . . . . . . . . . . . . . . . . . 291 4.4 Ricci Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 ∗Received December 12, 2005; accepted for publication April 16, 2006. †Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA ([email protected]). ‡Department of Mathematics, Zhongshan University, Guangzh ou 510275, P. R. China (stszxp@ zsu.edu.cn). 165 166 H.-D. CAO AND X.-P. ZHU 5 Long Time Behaviors 307 5.1 The Ricci Flow on Two-manifolds . . . . . . . . . . . . . . . . . . . . 3 08 5.2 Diﬀerentiable Sphere Theorems in 3-D and 4-D . . . . . . . . . . . . . 321 5.3 Nonsingular Solutions on Three-manifolds . . . . . . . . . . . . . . . . 336 6 Ancient κ-solutions 357 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 6.2 Asymptotic Shrinking Solitons . . . . . . . . . . . . . . . . . . . . . . 364 6.3 Curvature Estimates via Volume Growth . . . . . . . . . . . . . . . . . 373 6.4 Ancient κ-solutions on Three-manifolds . . . . . . . . . . . . . . . . . 384 7 Ricci Flow on Three-manifolds 398 7.1 Canonical Neighborhood Structures . . . . . . . . . . . . . . . . . . . 398 7.2 Curvature Estimates for Smooth Solutions . . . . . . . . . . . . . . . . 405 7.3 Ricci Flow with Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 7.4 Justiﬁcation of the Canonical Neighborhood Assumption s . . . . . . . 432 7.5 Curvature Estimates for Surgically Modiﬁed Solutions . . . . . . . . . 452 7.6 Long Time Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 7.7 Geometrization of Three-manifolds . . . . . . . . . . . . . . . . . . . . 481 References 486 Index 491 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 167 Introduction. In this paper, we shall present the Hamilton-Perelman theor y of Ricci ﬂow. Based on it, we shall give the ﬁrst written account of a complete proof of the Poincar´ e conjecture and the geometrization conject ure of Thurston. While the complete work is an accumulated eﬀorts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman. An important problem in diﬀerential geometry is to ﬁnd a cano nical metric on a given manifold. In turn, the existence of a canonical metri c often has profound topological implications. A good example is the classical u niformization theorem in two dimensions which, on one hand, provides a complete topol ogical classiﬁcation for compact surfaces, and on the other hand shows that every comp act surface has a canonical geometric structure: a metric of constant curvat ure. How to formulate and generalize this two-dimensional resul t to three and higher dimensional manifolds has been one of the most important and challenging topics in modern mathematics. In 1977, W. Thurston [122], based on ide as about Riemann sur- faces, Haken’s work and Mostow’s rigidity theorem, etc, for mulated a geometrization conjecture for three-manifolds which, roughly speaking, s tates that every compact ori- entable three-manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure. In particular, Thurston’ s conjecture contains, as a special case, the Poincar´ e conjecture: A closed three-man ifold with trivial fundamen- tal group is necessarily homeomorphic to the 3-sphere S3. In the past thirty years, many mathematicians have contributed to the understanding of this conjecture of Thurston. While Thurston’s theory is based on beautiful com bination of techniques from geometry and topology, there has been a powerful develo pment of geometric analysis in the past thirty years, lead by S.-T. Yau, R. Schoe n, C. Taubes, K. Uhlen- beck, and S. Donaldson, on the construction of canonical geo metric structures based on nonlinear PDEs (see, e.g., Yau’s survey papers [129, 130] ). Such canonical geo- metric structures include K¨ ahler-Einstein metrics, cons tant scalar curvature metrics, and self-dual metrics, among others. However, the most impo rtant contribution for geometric analysis on three-manifolds is due to Hamilton. In 1982, Hamilton [58] introduced the Ricci ﬂow ∂gij ∂t=−2Rij to study compact three-manifolds with positive Ricci curva ture. The Ricci ﬂow, which evolves a Riemannian metric by its Ricci curvature, is a natu ral analogue of the heat equation for metrics. As a consequence, the curvature tenso rs evolve by a system of diﬀusion equations which tends to distribute the curvature uniformly over the mani- fold. Hence, one expects that the initial metric should be im proved and evolve into a canonical metric, thereby leading to a better understandin g of the topology of the un- derlying manifold. In the celebrated paper [58], Hamilton s howed that on a compact three-manifold with an initial metric having positive Ricc i curvature, the Ricci ﬂow converges, after rescaling to keep constant volume, to a met ric of positive constant sectional curvature, proving the manifold is diﬀeomorphic to the three-sphere S3or a quotient of the three-sphere S3by a linear group of isometries. Shortly after, Yau sug- gested that the Ricci ﬂow should be the best way to prove the st ructure theorem for general three-manifolds. In the past two decades, Hamilton proved many important and remarkable theorems for the Ricci ﬂow, and laid the found ation for the program to approach the Poincar´ e conjecture and Thurston’s geomet rization conjecture via the Ricci ﬂow. 168 H.-D. CAO AND X.-P. ZHU The basic idea of Hamilton’s program can be brieﬂy described as follows. For any given compact three-manifold, one endows it with an arbitra ry (but can be suitably normalized by scaling) initial Riemannian metric on the man ifold and then studies the behavior of the solution to the Ricci ﬂow. If the Ricci ﬂow develops singularities, then one tries to ﬁnd out the structures of singularities so t hat one can perform (geometric) surgery by cutting oﬀ the singularities, and th en continue the Ricci ﬂow after the surgery. If the Ricci ﬂow develops singularities a gain, one repeats the process of performing surgery and continuing the Ricci ﬂow. If one ca n prove there are only a ﬁnite number of surgeries during any ﬁnite time interval and if the long-time behavior of solutions of the Ricci ﬂow with surgery is well understood , then one would recognize the topological structure of the initial manifold. Thus Hamilton’s program, when carried out successfully, wi ll give a proof of the Poincar´ e conjecture and Thurston’s geometrization conje cture. However, there were obstacles, most notably the veriﬁcation of the so called “Li ttle Loop Lemma” con- jectured by Hamilton [63] (see also [17]) which is a certain l ocal injectivity radius estimate, and the veriﬁcation of the discreteness of surger y times. In the fall of 2002 and the spring of 2003, Perelman [103, 104] brought in fresh n ew ideas to ﬁgure out important steps to overcome the main obstacles that remaine d in the program of Hamilton. (Indeed, in page 3 of [103], Perelman said “the imp lementation of Hamil- ton program would imply the geometrization conjecture for c losed three-manifolds” and “In this paper we carry out some details of Hamilton progr am”.) Perelman’s breakthrough on the Ricci ﬂow excited the entire mathematic s community. His work has since been examined to see whether the proof of the Poinca r´ e conjecture and geometrization program, based on the combination of Hamilt on’s fundamental ideas and Perelman’s new ideas, holds together. The present paper grew out of such an eﬀort. Now we describe the three main parts of Hamilton’s program in more detail. (i) Determine the structures of singularities Given any compact three-manifold Mwith an arbitrary Riemannian metric, one evolves the metric by the Ricci ﬂow. Then, as Hamilton showed in [58], the solution g(t) to the Ricci ﬂow exists for a short time and is unique (also se e Theorem 1.2.1). In fact, Hamilton [58] showed that the solution g(t) will exist on a maximal time interval [0,T), where either T=∞, or 0< T < ∞and the curvature becomes unbounded asttends toT. We call such a solution g(t) a maximal solution of the Ricci ﬂow. If T <∞and the curvature becomes unbounded as ttends toT, we say the maximal solution develops singularities as ttends toTandTis the singular time. In the early 1990s, Hamilton systematically developed meth ods to understand the structure of singularities. In [61], based on suggestion by Yau, he proved the funda- mental Li-Yau [82] type diﬀerential Harnack estimate (the L i-Yau-Hamilton estimate) for the Ricci ﬂow with nonnegative curvature operator in all dimensions. With the help of Shi’s interior derivative estimate [114], he [62] es tablished a compactness the- orem for smooth solutions to the Ricci ﬂow with uniformly bou nded curvatures and uniformly bounded injectivity radii at the marked points. B y imposing an injectivity radius condition, he rescaled the solution to show that each singularity is asymptotic to one of the three types of singularity models [63]. In [63] h e discovered (also inde- pendently by Ivey [73]) an amazing curvature pinching estim ate for the Ricci ﬂow on three-manifolds. This pinching estimate implies that any t hree-dimensional singular- ity model must have nonnegative curvature. Thus in dimensio n three, one only needs to obtain a complete classiﬁcation for nonnegatively curve d singularity models. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 169 For Type I singularities in dimension three, Hamilton [63] e stablished an isoperi- metric ratio estimate to verify the injectivity radius cond ition and obtained spherical or necklike structures for any Type I singularity model. Bas ed on the Li-Yau-Hamilton estimate, he showed that any Type II singularity model with n onnegative curvature is either a steady Ricci soliton with positive sectional cur vature or the product of the so called cigar soliton with the real line [66]. (Charact erization for nonnegatively curved Type III models was obtained in [30].) Furthermore, h e developed a dimension reduction argument to understand the geometry of steady Ric ci solitons [63]. In the three-dimensional case, he showed that each steady Ricci so liton with positive curva- ture has some necklike structure. Hence Hamilton had basica lly obtained a canonical neighborhood structure at points where the curvature is com parable to the maximal curvature for solutions to the three-dimensional Ricci ﬂow . However two obstacles remained: (a) the veriﬁcation of the i mposed injectivity radius condition in general; and (b) the possibility of form ing a singularity modelled on the product of the cigar soliton with a real line which coul d not be removed by surgery. The recent spectacular work of Perelman [103] remo ved these obstacles by establishing a local injectivity radius estimate, which is valid for the Ricci ﬂow on compact manifolds in all dimensions. More precisely, Perel man proved two versions of “no local collapsing” property (Theorem 3.3.3 and Theore m 3.3.2), one with an entropy functional he introduced in [103], which is monoton e under the Ricci ﬂow, and the other with a space-time distance function obtained b y path integral, analogous to what Li-Yau did in [82], which gives rise to a monotone volu me-type (called reduced volume by Perelman) estimate. By combining Perelman’s no lo cal collapsing theorem I′(Theorem 3.3.3) with the injectivity radius estimate of Che ng-Li-Yau (Theorem 4.2.2), one immediately obtains the desired injectivity ra dius estimate, or the Little Loop Lemma (Theorem 4.2.4) conjectured by Hamilton. Furthermore, Perelman [103] developed a reﬁned rescaling a rgument (by consider- ing local limits and weak limits in Alexandrov spaces) for si ngularities of the Ricci ﬂow on three-manifolds to obtain a uniform and global version of the canonical neighbor- hood structure theorem. We would like to point out that our pr oof of the singularity structure theorem (Theorem 7.1.1) is diﬀerent from that of P erelman in two aspects: (1) we avoid using his crucial estimate in Claim 2 in Section 1 2.1 of [103]; (2) we give a new approach to extend the limit backward in time to an ancie nt solution. These diﬀerences are due to the diﬃculties in understanding Perel man’s arguments at these points. (ii) Geometric surgeries and the discreteness of surgery ti mes After obtaining the canonical neighborhoods (consisting o f spherical, necklike and caplike regions) for the singularities, one would like to pe rform geometric surgery and then continue the Ricci ﬂow. In [64], Hamilton initiated suc h a surgery procedure for the Ricci ﬂow on four-manifolds with positive isotropic curvature and presented a concrete method for performing the geometric surgery. His surgery procedures can be roughly described as follows: cutting the neck-like regi ons, gluing back caps, and removing the spherical regions. As will be seen in Section 7. 3 of this paper, Hamilton’s geometric surgery method also works for the Ricci ﬂow on comp act orientable three- manifolds. Now an important challenge is to prevent surgery times from a ccumulating and make sure one performs only a ﬁnite number of surgeries on eac h ﬁnite time interval. The problem is that, when one performs the surgeries with a gi ven accuracy at each surgery time, it is possible that the errors may add up to a cer tain amount which 170 H.-D. CAO AND X.-P. ZHU could cause the surgery times to accumulate. To prevent this from happening, as time goes on, successive surgeries must be performed with in creasing accuracy. In [104], Perelman introduced some brilliant ideas which allo w one to ﬁnd “ﬁne” necks, glue “ﬁne” caps, and use rescaling to prove that the surgery t imes are discrete. When using the rescaling argument for surgically modiﬁed so lutions of the Ricci ﬂow, one encounters the diﬃculty of how to apply Hamilton’s c ompactness theorem (Theorem 4.1.5), which works only for smooth solutions. The idea to overcome this diﬃculty consists of two parts. The ﬁrst part, due to Perelma n [104], is to choose the cutoﬀ radius in neck-like regions small enough to push the su rgical regions far away in space. The second part, due to the authors and Chen-Zhu [34 ], is to show that the surgically modiﬁed solutions are smooth on some uniform (sm all) time intervals (on compact subsets) so that Hamilton’s compactness theorem ca n still be applied. To do so, we establish three time-extension results (see Step 2 in the proof of Proposition 7.4.1.). Perhaps, this second part is more crucial. Without it, Shi’s interior derivative estimate (Theorem 1.4.2) may not applicable, and hence one c annot be certain that Hamilton’s compactness theorem holds when only having the u niformC0bound on curvatures. We remark that in our proof of this second part, a s can be seen in the proof of Proposition 7.4.1, we require a deep comprehension of the prolongation of the gluing “ﬁne” caps for which we will use the recent uniquen ess theorem of Bing- Long Chen and the second author [33] for solutions of the Ricc i ﬂow on noncompact manifolds. Once surgeries are known to be discrete in time, one can compl ete the classiﬁca- tion, started by Schoen-Yau [109, 110], for compact orienta ble three-manifolds with positive scalar curvature. More importantly, for simply co nnected three-manifolds, if one can show that solutions to the Ricci ﬂow with surgery beco me extinct in ﬁnite time, then the Poincar´ e conjecture would follow. Such a ﬁni te extinction time re- sult was proposed by Perelman [105], and a proof also appears in Colding-Minicozzi [42]. Thus, the combination of Theorem 7.4.3 (i) and the ﬁnit e extinction time result provides a complete proof to the Poincar´ e conjecture. (iii) The long-time behavior of surgically modiﬁed solutio ns. To approach the structure theorem for general three-manifo lds, one still needs to analyze the long-time behavior of surgically modiﬁed sol utions to the Ricci ﬂow. In [65], Hamilton studied the long time behavior of the Ricci ﬂow on compact three- manifolds for a special class of (smooth) solutions, the so c alled nonsingular solutions. These are the solutions that, after rescaling to keep consta nt volume, have (uniformly) bounded curvature for all time. Hamilton [65] proved that an y three-dimensional non- singular solution either collapses or subsequently conver ges to a metric of constant curvature on the compact manifold or, at large time, admits a thick-thin decompo- sition where the thick part consists of a ﬁnite number of hype rbolic pieces and the thin part collapses. Moreover, by adapting Schoen-Yau’s mi nimal surface arguments in [110] and using a result of Meeks-Yau [86], Hamilton showe d that the boundary of hyperbolic pieces are incompressible tori. Consequentl y, when combined with the collapsing results of Cheeger-Gromov [24, 25], this shows t hat any nonsingular solu- tion to the Ricci ﬂow is geometrizable in the sense of Thursto n [122]. Even though the nonsingular assumption seems very restrictive and ther e are few conditions known so far which can guarantee a solution to be nonsingular, neve rtheless the ideas and arguments of Hamilton’s work [65] are extremely important. In [104], Perelman modiﬁed Hamilton’s arguments to analyze the long-time be- THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 171 havior of arbitrary smooth solutions to the Ricci ﬂow and sol utions with surgery to the Ricci ﬂow in dimension three. Perelman also argued that t he proof of Thurston’s geometrization conjecture could be based on a thick-thin de composition, but he could only show the thin part will only have a (local) lower bound on the sectional cur- vature. For the thick part, based on the Li-Yau-Hamilton est imate, Perelman [104] established a crucial elliptic type estimate, which allowe d him to conclude that the thick part consists of hyperbolic pieces. For the thin part, he announced in [104] a new collapsing result which states that if a three-manifol d collapses with (local) lower bound on the sectional curvature, then it is a graph man ifold. Assuming this new collapsing result, Perelman [104] claimed that the solu tions to the Ricci ﬂow with surgery have the same long-time behavior as nonsingula r solutions in Hamilton’s work, a conclusion which would imply a proof of Thurston’s ge ometrization conjec- ture. Although the proof of this new collapsing result promi sed by Perelman in [104] is still not available in literature, Shioya-Yamaguchi [11 8] has published a proof of the collapsing result in the special case when the manifold is cl osed. In the last section of this paper (see Theorem 7.7.1), we will provide a proof of T hurston’s geometriza- tion conjecture by only using Shioya-Yamaguchi’s collapsi ng result. In particular, this gives another proof of the Poincar´ e conjecture. We would like to point out that Perelman [104] did not quite gi ve an explicit statement of the thick-thin decomposition for surgical sol utions. When we were trying to write down an explicit statement, we needed to add a restri ction on the relation between the accuracy parameter εand the collapsing parameter w. Nevertheless, we are still able to obtain a weaker version of the thick-thin de composition (Theorem 7.6.3) that is suﬃcient to deduce the geometrization result . In this paper, we shall give complete and detailed proofs of w hat we outlined above, especially of Perelman’s work in his second paper [10 4] in which many key ideas of the proofs are sketched or outlined but complete det ails of the proofs are often missing. As we pointed out before, we have to substitut e several key arguments of Perelman by new approaches based on our study, because we w ere unable to com- prehend these original arguments of Perelman which are esse ntial to the completion of the geometrization program. Our paper is aimed at both graduate students and researchers who want to learn Hamilton’s Ricci ﬂow and to understand the Hamilton-Perelm an theory and its appli- cation to the geometrization of three-manifolds. For this p urpose, we have made the paper to be essentially self-contained so that the proof of t he geometrization is acces- sible to those who are familiar with basics of Riemannian geo metry and elliptic and parabolic partial diﬀerential equations. The reader may ﬁn d some original papers, particularly those of Hamilton’s on the Ricci ﬂow, before th e appearance of Perel- man’s preprints in the book “Collected Papers on Ricci Flow” [17]. For introductory materials to the Hamilton-Perelman theory of Ricci ﬂow, we a lso refer the reader to the recent book by B. Chow and D. Knopf [39] and the forthcomin g book by B. Chow, P. Lu and L. Ni [41]. We remark that there have also appeared se veral sets of notes on Perelman’s work, including the one written by B. Kleiner a nd J. Lott [78], which cover part of the materials that are needed for the geometriz ation program. There also have appeared several survey articles by Cao-Chow [16] , Milnor [91], Anderson [4] and Morgan [95] for the geometrization of three-manifol ds via the Ricci ﬂow. We are very grateful to Professor S.-T. Yau, who suggested us to write this paper based on our notes, for introducing us to the wonderland of th e Ricci ﬂow. His vision and strong belief in the Ricci ﬂow encouraged us to per severe. We also thank 172 H.-D. CAO AND X.-P. ZHU him for his many suggestions and constant encouragement. Wi thout him, it would be impossible for us to ﬁnish this paper. We are enormously in debted to Professor Richard Hamilton for creating the Ricci ﬂow and developing t he entire program to approach the geometrization of three-manifolds. His work o n the Ricci ﬂow and other geometric ﬂows has inﬂuenced on virtually everyone in the ﬁe ld. The ﬁrst author especially would like to thank Professor Hamilton for teach ing him so much about the subject over the past twenty years, and for his constant enco uragement and friendship. We are indebted to Dr. Bing-Long Chen, who contributed a grea t deal in the process of writing this paper. We beneﬁted a lot from constan t discussions with him on the subjects of geometric ﬂows and geometric analysis. He also contributed many ideas in various proofs in the paper. We would like to thank Ms . Huiling Gu, a Ph.D student of the second author, for spending many months of goi ng through the entire paper and checking the proofs. Without both of them, it would take much longer time for us to ﬁnish this paper. The ﬁrst author would like to express his gratitude to the Joh n Simon Guggen- heim Memorial Foundation, the National Science Foundation (grants DMS-0354621 and DMS-0506084), and the Outstanding Overseas Young Schol ar Fund of Chinese National Science Foundation for their support for the resea rch in this paper. He also would like to thank Tsinghua University in Beijing for its ho spitality and support while he was working there. The second author wishes to thank his wife, Danlin Liu, for her understanding and support over all these years. The s econd author is also indebted to the National Science Foundation of China for the support in his work on geometric ﬂows, some of which has been incorporated in this p aper. The last part of the work in this paper was done and the material in Chapter 3, C hapter 6 and Chap- ter 7 was presented while the second author was visiting the H arvard Mathematics Department in the fall semester of 2005 and the early spring s emester of 2006. He wants to especially thank Professor Shing-Tung Yau, Profes sor Cliﬀ Taubes and Pro- fessor Daniel W. Stroock for the enlightening comments and e ncouragement during the lectures. Also he gratefully acknowledges the hospital ity and the ﬁnancial support of Harvard University. 1. Evolution Equations. In this chapter, we introduce Hamilton’s Ricci ﬂow and derive evolution equations of curvatures. The short tim e existence and uniqueness theorem of the Ricci ﬂow on a compact manifold is proved in Sec tion 1.2. In Section 1.4, we prove Shi’s local derivative estimate, which plays a n important role in the Ricci ﬂow. Perelman’s two functionals and their monotonicity pro perties are discussed in Section 1.5. 1.1. The Ricci Flow. LetMbe ann-dimensional complete Riemannian man- ifold with the Riemannian metric gij. The Levi-Civita connection is given by the Christoﬀel symbols Γk ij=1 2gkl/parenleftbigg∂gjl ∂xi+∂gil ∂xj−∂gij ∂xl/parenrightbigg wheregijis the inverse of gij. The summation convention of summing over repeated indices is used here and throughout the book. The Riemannian curvature tensor is given by Rk ijl=∂Γk jl ∂xi−∂Γk il ∂xj+ Γk ipΓp jl−Γk jpΓp il. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 173 We lower the index to the third position, so that Rijkl=gkpRp ijl. The curvature tensor Rijklis anti-symmetric in the pairs i, jandk, land symmetric in their interchange: Rijkl=−Rjikl=−Rijlk=Rklij. Also the ﬁrst Bianchi identity holds (1.1.1) Rijkl+Rjkil+Rkijl= 0. The Ricci tensor is the contraction Rik=gjlRijkl, and the scalar curvature is R=gijRij. We denote the covariant derivative of a vector ﬁeld v=vj∂ ∂xjby ∇ivj=∂vj ∂xi+ Γj ikvk and of a 1-form by ∇ivj=∂vj ∂xi−Γk ijvk. These deﬁnitions extend uniquely to tensors so as to preserv e the product rule and contractions. For the exchange of two covariant derivative s, we have ∇i∇jvl− ∇ j∇ivl=Rl ijkvk, (1.1.2) ∇i∇jvk− ∇ j∇ivk=Rijklglmvm, (1.1.3) and similar formulas for more complicated tensors. The seco nd Bianchi identity is given by (1.1.4) ∇mRijkl+∇iRjmkl+∇jRmikl= 0. For any tensor T=Ti jkwe deﬁne its length by |Ti jk|2=gilgjmgkpTi jkTl mp, and we deﬁne its Laplacian by ∆Ti jk=gpq∇p∇qTi jk, the trace of the second iterated covariant derivatives. Sim ilar deﬁnitions hold for more general tensors. TheRicci ﬂow of Hamilton [58] is the evolution equation (1.1.5)∂gij ∂t=−2Rij 174 H.-D. CAO AND X.-P. ZHU for a family of Riemannian metrics gij(t) onM. It is a nonlinear system of second order partial diﬀerential equations on metrics. In order to get a feel for the Ricci ﬂow (1.1.5) we ﬁrst present some examples of speciﬁc solutions. (1) Einstein metrics A Riemannian metric gijis called Einstein if Rij=λgij for some constant λ. A smooth manifold Mwith an Einstein metric is called an Einstein manifold . If the initial metric is Ricci ﬂat, so that Rij= 0, then clearly the metric does not change under (1.1.5). Hence any Ricci ﬂat metric is a stat ionary solution of the Ricci ﬂow. This happens, for example, on a ﬂat torus or on any K3-surface with a Calabi-Yau metric. If the initial metric is Einstein with positive scalar curva ture, then the metric will shrink under the Ricci ﬂow by a time-dependent factor. I ndeed, since the initial metric is Einstein, we have Rij(x,0) =λgij(x,0),∀x∈M and someλ>0. Let gij(x,t) =ρ2(t)gij(x,0). From the deﬁnition of the Ricci tensor, one sees that Rij(x,t) =Rij(x,0) =λgij(x,0). Thus the equation (1.1.5) corresponds to ∂(ρ2(t)gij(x,0)) ∂t=−2λgij(x,0). This gives the ODE (1.1.6)dρ dt=−λ ρ, whose solution is given by ρ2(t) = 1−2λt. Thus the evolving metric gij(x,t) shrinks homothetically to a point as t→T= 1/2λ. Note that as t→T, the scalar curvature becomes inﬁnite like 1 /(T−t). By contrast, if the initial metric is an Einstein metric of ne gative scalar curvature, the metric will expand homothetically for all times. Indeed if Rij(x,0) =−λgij(x,0) withλ>0 and gij(x,t) =ρ2(t)gij(x,0). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 175 Thenρ(t) satisﬁes the ODE (1.1.7)dρ dt=λ ρ, with the solution ρ2(t) = 1 + 2λt. Hence the evolving metric gij(x,t) =ρ2(t)gij(x,0) exists and expands homothetically for all times, and the curvature will fall back to zero like −1/t. Note that now the evolving metric gij(x,t) only goes back in time to −1/2λ, when the metric explodes out of a single point in a “big bang”. (2) Ricci Solitons We will call a solution to an evolution equation which moves u nder a one- parameter subgroup of the symmetry group of the equation a steady soliton . The symmetry group of the Ricci ﬂow contains the full diﬀeomorph ism group. Thus a solution to the Ricci ﬂow (1.1.5) which moves by a one-parame ter group of diﬀeomor- phismsϕtis called a steady Ricci soliton . Ifϕtis a one-parameter group of diﬀeomorphisms generated by a ve ctor ﬁeldV onM, then the Ricci soliton is given by (1.1.8) gij(x,t) =ϕ∗ tgij(x,0) which implies that the Ricci term −2Ricon the RHS of (1.1.5) is equal to the Lie derivative LVgof the evolving metric g. In particular, the initial metric gij(x,0) satisﬁes the following steady Ricci soliton equation (1.1.9) 2 Rij+gik∇jVk+gjk∇iVk= 0. If the vector ﬁeld Vis the gradient of a function fthen the soliton is called a steady gradient Ricci soliton . Thus (1.1.10) Rij+∇i∇jf= 0,orRic+∇2f= 0, is the steady gradient Ricci soliton equation. Conversely, it is clear that a metric gijsatisfying (1.1.10) generates a steady gradient Ricci soliton gij(t) given by (1.1.8). For this reason we also often call such a metricgija steady gradient Ricci soliton and do not necessarily disti nguish it with the solution gij(t) it generates. More generally, we can consider a solution to the Ricci ﬂow (1 .1.5) which moves by diﬀeomorphisms and also shrinks or expands by a (time-dep endent) factor at the same time. Such a solution is called a homothetically shrinking or homothetically expanding Ricci soliton . The equation for a homothetic Ricci soliton is (1.1.11) 2 Rij+gik∇jVk+gjk∇iVk−2λgij= 0, or for a homothetic gradient Ricci soliton, (1.1.12) Rij+∇i∇jf−λgij= 0, whereλis the homothetic constant. For λ >0 the soliton is shrinking, for λ <0 it is expanding. The case λ= 0 is a steady Ricci soliton, the case V= 0 (orfbeing 176 H.-D. CAO AND X.-P. ZHU a constant function) is an Einstein metric. Thus Ricci solit ons can be considered as natural extensions of Einstein metrics. In fact, the foll owing result states that there are no nontrivial gradient steady or expanding Ricci s olitons on any compact manifold. We remark that if the underlying manifold Mis a complex manifold and the initial metric is K¨ ahler, then it is well known (see, e.g., [ 62, 11]) that the solution metric to the Ricci ﬂow (1.1.5) remains K¨ ahler. For this rea son, the Ricci ﬂow on a K¨ ahler manifold is called the K¨ ahler-Ricci ﬂow . A (steady, or shrinking, or expanding) Ricci soliton to the K¨ ahler-Ricci ﬂow is called a (steady , orshrinking , orexpanding repectively) K¨ ahler-Ricci soliton . Proposition 1.1.1. On a compact n-dimensional manifold M, a gradient steady or expanding Ricci soliton is necessarily an Einstein metri c. Proof. We shall only prove the steady case and leave the expanding ca se as an exercise. Our argument here follows that of Hamilton [63]. Letgijbe a complete steady gradient Ricci soliton on a manifold Mso that Rij+∇i∇jf= 0. Taking the trace, we get (1.1.13) R+ ∆f= 0. Also, taking the covariant derivatives of the Ricci soliton equation, we have ∇i∇j∇kf− ∇ j∇i∇kf=∇jRik− ∇ iRjk. On the other hand, by using the commutating formula (1.1.3), we otain ∇i∇j∇kf− ∇ j∇i∇kf=Rijkl∇lf. Thus ∇iRjk− ∇ jRik+Rijkl∇lf= 0. Taking the trace on jandk, and using the contracted second Bianchi identity (1.1.14) ∇jRij=1 2∇iR, we get ∇iR−2Rij∇jf= 0. Then ∇i(|∇f|2+R) = 2∇jf(∇i∇jf+Rij) = 0. Therefore (1.1.15) R+|∇f|2=C for some constant C. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 177 Taking the diﬀerence of (1.1.13) and (1.1.15), we get (1.1.16) ∆ f− |∇f|2=−C. We claimC= 0 whenMis compact. Indeed, this follows either from (1.1.17) 0 = −/integraldisplay M∆(e−f)dV=/integraldisplay M(∆f− |∇f|2)e−fdV, or from considering (1.1.16) at both the maximum point and mi nimum point of f. Then, by integrating (1.1.16) we obtain /integraldisplay M|∇f|2dV= 0. Thereforefis a constant and gijis Ricci ﬂat. Remark 1.1.2. By contrast, there do exist nontrivial compact gradient shr inking Ricci solitons (see Koiso [80], Cao [13] and Wang-Zhu [127] ) . Also, there exist complete noncompact steady gradient Ricci solitons that ar e not Ricci ﬂat. In two dimensions Hamilton [60] wrote down the ﬁrst such example on R2, called the cigar soliton , where the metric is given by (1.1.18) ds2=dx2+dy2 1 +x2+y2, and the vector ﬁeld is radial, given by V=−∂/∂r =−(x∂/∂x +y∂/∂y ).This metric has positive curvature and is asymptotic to a cylinde r of ﬁnite circumference 2πat∞. Higher dimensional examples were found by Robert Bryant [1 0] on Rnin the Riemannian case, and by the ﬁrst author [13] on Cnin the K¨ ahler case. These examples are complete, rotationally symmetric, of positiv e curvature and found by solving certain nonlinear ODE (system). Noncompact expand ing solitons were also constructed by the ﬁrst author [13]. More recently, Feldman , Ilmanen and Knopf [46] constructed new examples of noncompact shrinking and e xpanding K¨ ahler-Ricci solitons. 1.2. Short-time Existence and Uniqueness. In this section we establish the short-time existence and uniqueness result for the Ricci ﬂo w (1.1.5) on a compact n- dimensional manifold M. We will see that the Ricci ﬂow is a system of second order nonlinear weakly parabolic partial diﬀerential equations . In fact, the degeneracy of the system is caused by the diﬀeomorphism group of Mwhich acts as the gauge group of the Ricci ﬂow. For any diﬀeomorphism ϕofM, we have Ric ( ϕ∗(g)) =ϕ∗(Ric (g)). Thus, ifg(t) is a solution to the Ricci ﬂow (1.1.5), so is ϕ∗(g(t)). Because the Ricci ﬂow (1.1.5) is only weakly parabolic, even the existence and uniqueness result on a compact manifold does not follow from standard PDE theory. The short-time existence and uniqueness result in the compa ct case is ﬁrst proved by Hamilton [58] using the Nash-Moser implicit function theor em. Shortly after Denis De Turck [43] gave a much simpler proof using the gauge ﬁxing i dea which we will present here. In the noncompact case, the short-time existence was establ ished by Shi [114] in 1989, but the uniqueness result has been proved only very rec ently by Bing-Long Chen and the second author. These results will be presented at the end of this section. 178 H.-D. CAO AND X.-P. ZHU LetMbe a compact n-dimensional Riemannian manifold. The Ricci ﬂow equation is a second order nonlinear partial diﬀerential system (1.2.1)∂ ∂tgij=E(gij), for a family of Riemannian metrics gij(·,t) onM, where E(gij) =−2Rij =−2/parenleftbigg∂ ∂xkΓk ij−∂ ∂xiΓk kj+ Γk kpΓp ij−Γk ipΓp kj/parenrightbigg =∂ ∂xi/braceleftbigg gkl∂ ∂xjgkl/bracerightbigg −∂ ∂xk/braceleftbigg gkl/parenleftbigg∂ ∂xigjl+∂ ∂xjgil−∂ ∂xlgij/parenrightbigg/bracerightbigg + 2Γk ipΓp kj−2Γk kpΓp ij. The linearization of this system is ∂˜gij ∂t=DE(gij)˜gij where ˜gijis the variation in gijandDEis the derivative of Egiven by DE(gij)˜gij=gkl/braceleftbigg∂2˜gkl ∂xi∂xj−∂2˜gjl ∂xi∂xk−∂2˜gil ∂xj∂xk+∂2˜gij ∂xk∂xl/bracerightbigg + (lower order terms). We now compute the symbol of DE. This is to take the highest order derivatives and replace∂ ∂xiby the Fourier transform variable ζi. The symbol of the linear diﬀerential operatorDE(gij) in the direction ζ= (ζ1,...,ζ n) is σDE(gij)(ζ)˜gij=gkl(ζiζj˜gkl+ζkζl˜gij−ζiζk˜gjl−ζjζk˜gil). To see what the symbol does, we can always assume ζhas length 1 and choose coordinates at a point such that   gij=δij, ζ= (1,0,...,0). Then (σDE(gij)(ζ))(˜gij) = ˜gij+δi1δj1(˜g11+···+ ˜gnn) −δi1˜g1j−δj1˜g1i, i.e., [σDE(gij)(ζ)(˜gij)]11= ˜g22+···+ ˜gnn, [σDE(gij)(ζ)(˜gij)]1k= 0,ifk/\e}atio\slash= 1, [σDE(gij)(ζ)(˜gij)]kl= ˜gkl,ifk/\e}atio\slash= 1,l/\e}atio\slash= 1. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 179 In particular (˜gij) = ∗ ∗ ··· ∗ ∗0···0 ............ ∗0···0  are zero eigenvectors of the symbol. The presence of the zero eigenvalue shows that the system can not be strictly parabolic. Therefore, instead of considering the system (1 .2.1) (or the Ricci ﬂow equation (1.1.5)) we will follow a trick of De Turck[43] to co nsider a modiﬁed evolution equation, which turns out to be strictly parabolic, so that w e can apply the standard theory of parabolic equations. Suppose ˆgij(x,t) is a solution of the Ricci ﬂow (1.1.5), and ϕt:M→Mis a family of diﬀeomorphisms of M. Let gij(x,t) =ϕ∗ tˆgij(x,t) be the pull-back metrics. We now want to ﬁnd the evolution equ ation for the metrics gij(x,t). Denote by y(x,t) =ϕt(x) ={y1(x,t),y2(x,t),...,yn(x,t)} in local coordinates. Then (1.2.2) gij(x,t) =∂yα ∂xi∂yβ ∂xjˆgαβ(y,t), and ∂ ∂tgij(x,t) =∂ ∂t/bracketleftbigg∂yα ∂xi∂yβ ∂xjˆgαβ(y,t)/bracketrightbigg =∂yα ∂xi∂yβ ∂xj∂ ∂tˆgαβ(y,t) +∂ ∂xi/parenleftbigg∂yα ∂t/parenrightbigg∂yβ ∂xjˆgαβ(y,t) +∂yα ∂xi∂ ∂xj/parenleftbigg∂yβ ∂t/parenrightbigg ˆgαβ(y,t). Let us choose a normal coordinate {xi}around a ﬁxed point p∈Msuch that∂gij ∂xk= 0 atp. Since ∂ ∂tˆgαβ(y,t) =−2ˆRαβ(y,t) +∂ˆgαβ ∂yγ∂yγ ∂t, 180 H.-D. CAO AND X.-P. ZHU we have in the normal coordinate, ∂ ∂tgij(x,t) =−2∂yα ∂xi∂yβ ∂xjˆRαβ(y,t) +∂yα ∂xi∂yβ ∂xj∂ˆgαβ ∂yγ∂yγ ∂t +∂ ∂xi/parenleftbigg∂yα ∂t/parenrightbigg∂yβ ∂xjˆgαβ(y,t) +∂ ∂xj/parenleftbigg∂yβ ∂t/parenrightbigg∂yα ∂xiˆgαβ(y,t) =−2Rij(x,t) +∂yα ∂xi∂yβ ∂xj∂ˆgαβ ∂yγ∂yγ ∂t+∂ ∂xi/parenleftbigg∂yα ∂t/parenrightbigg∂xk ∂yαgjk +∂ ∂xj/parenleftbigg∂yβ ∂t/parenrightbigg∂xk ∂yβgik =−2Rij(x,t) +∂yα ∂xi∂yβ ∂xj∂ˆgαβ ∂yγ∂yγ ∂t+∂ ∂xi/parenleftbigg∂yα ∂t∂xk ∂yαgjk/parenrightbigg +∂ ∂xj/parenleftbigg∂yβ ∂t∂xk ∂yβgik/parenrightbigg −∂yα ∂t∂ ∂xi/parenleftbigg∂xk ∂yα/parenrightbigg gjk−∂yβ ∂t∂ ∂xj/parenleftbigg∂xk ∂yβ/parenrightbigg gik. The second term on the RHS gives, in the normal coordinate, ∂yα ∂xi∂yβ ∂xj∂yγ ∂t∂ˆgαβ ∂yγ=∂yα ∂xi∂yβ ∂xj∂yγ ∂tgkl∂ ∂yγ/parenleftbigg∂xk ∂yα∂xl ∂yβ/parenrightbigg =∂yα ∂xi∂yγ ∂t∂ ∂yγ/parenleftbigg∂xk ∂yα/parenrightbigg gjk+∂yβ ∂xj∂yγ ∂t∂ ∂yγ/parenleftbigg∂xk ∂yβ/parenrightbigg gik =∂yα ∂t∂2xk ∂yα∂yβ∂yβ ∂xigjk+∂yβ ∂t∂2xk ∂yα∂yβ∂yα ∂xjgik =∂yα ∂t∂ ∂xi/parenleftbigg∂xk ∂yα/parenrightbigg gjk+∂yβ ∂t∂ ∂xj/parenleftbigg∂xk ∂yβ/parenrightbigg gik. So we get ∂ ∂tgij(x,t) (1.2.3) =−2Rij(x,t) +∇i/parenleftbigg∂yα ∂t∂xk ∂yαgjk/parenrightbigg +∇j/parenleftbigg∂yβ ∂t∂xk ∂yβgik/parenrightbigg . If we deﬁne y(x,t) =ϕt(x) by the equations (1.2.4)  ∂yα ∂t=∂yα ∂xkgjl(Γk jl−o Γk jl), yα(x,0) =xα, andVi=gikgjl(Γk jl−o Γk jl), we get the following evolution equation for the pull-back metric (1.2.5)  ∂ ∂tgij(x,t) =−2Rij(x,t) +∇iVj+∇jVi, gij(x,0) =ogij(x), THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 181 whereogij(x) is the initial metric ando Γk jlis the connection of the initial metric. Lemma 1.2.1. The modiﬁed evolution equation (1.2.5)is a strictly parabolic system. Proof. The RHS of the equation (1.2.5) is given by −2Rij(x,t) +∇iVj+∇jVi =∂ ∂xi/braceleftbigg gkl∂gkl ∂xj/bracerightbigg −∂ ∂xk/braceleftbigg gkl/parenleftbigg∂gjl ∂xi+∂gil ∂xj−∂gij ∂xl/parenrightbigg/bracerightbigg +gjkgpq∂ ∂xi/braceleftbigg1 2gkl/parenleftbigg∂gpl ∂xq+∂gql ∂xp−∂gpq ∂xl/parenrightbigg/bracerightbigg +gikgpq∂ ∂xj/braceleftbigg1 2gkl/parenleftbigg∂gpl ∂xq+∂gql ∂xp−∂gpq ∂xl/parenrightbigg/bracerightbigg + (lower order terms) =gkl/braceleftbigg∂2gkl ∂xi∂xj−∂2gjl ∂xi∂xk−∂2gil ∂xj∂xk+∂2gij ∂xk∂xl/bracerightbigg +1 2gpq/braceleftbigg∂2gpj ∂xi∂xq+∂2gqj ∂xi∂xp−∂2gpq ∂xi∂xj/bracerightbigg +1 2gpq/braceleftbigg∂2gpi ∂xj∂xq+∂2gqi ∂xj∂xp−∂2gpq ∂xi∂xj/bracerightbigg + (lower order terms) =gkl∂2gij ∂xk∂xl+ (lower order terms) . Thus its symbol is ( gklζkζl)˜gij. Hence the equation in (1.2.5) is strictly parabolic. Now since the equation (1.2.5) is strictly parabolic and the manifoldMis compact, it follows from the standard theory of parabolic equations ( see for example [81]) that (1.2.5) has a solution for a short time. From the solution of ( 1.2.5) we can obtain a solution of the Ricci ﬂow from (1.2.4) and (1.2.2). This sho ws existence. Now we argue the uniqueness of the solution. Since Γk jl=∂yα ∂xj∂yβ ∂xl∂xk ∂yγˆΓγ αβ+∂xk ∂yα∂2yα ∂xj∂xl, the initial value problem (1.2.4) can be written as (1.2.6)  ∂yα ∂t=gjl/parenleftbigg ∂2yα ∂xj∂xl−o Γk jl∂yα ∂xk+ˆΓα γβ∂yβ ∂xj∂yγ ∂xl/parenrightbigg , yα(x,0) =xα. This is clearly a strictly parabolic system. For any two solu tions ˆg(1) ij(·,t) and ˆg(2) ij(·,t) of the Ricci ﬂow (1.1.5) with the same initial data, we can sol ve the initial value problem (1.2.6) (or equivalently, (1.2.4)) to get two famil iesϕ(1) tandϕ(2) tof dif- feomorphisms of M. Thus we get two solutions, g(1) ij(·,t) = (ϕ(1) t)∗ˆg(1) ij(·,t) and g(2) ij(·,t) = (ϕ(2) t)∗ˆg(2) ij(·,t), to the modiﬁed evolution equation (1.2.5) with the same 182 H.-D. CAO AND X.-P. ZHU initial metric. The uniqueness result for the strictly para bolic equation (1.2.5) implies thatg(1) ij=g(2) ij. Then by equation (1.2.4) and the standard uniqueness resul t of ODE systems, the corresponding solutions ϕ(1) tandϕ(2) tof (1.2.4) (or equivalently (1.2.6)) must agree. Consequently the metrics ˆ g(1) ijand ˆg(2) ijmust agree also. Thus we have proved the following result. Theorem 1.2.2 ( Hamilton [58], De Turck [43] ).Let(M, g ij(x))be a compact Riemannian manifold. Then there exists a constant T >0such that the initial value problem   ∂ ∂tgij(x,t) =−2Rij(x,t) gij(x,0) =gij(x) has a unique smooth solution gij(x,t)onM×[0,T). The case of a noncompact manifold is much more complicated an d involves a huge amount of techniques from the theory of partial diﬀeren tial equations. Here we will only state the existence and uniqueness results and ref er the reader to the cited references for the proofs. The following existence result was obtained by Shi [114] in h is thesis published in 1989. Theorem 1.2.3 ( Shi [114] ).Let(M, g ij(x))be a complete noncompact Rie- mannian manifold of dimension nwith bounded curvature. Then there exists a con- stantT >0such that the initial value problem   ∂ ∂tgij(x,t) =−2Rij(x,t) gij(x,0) =gij(x) has a smooth solution gij(x,t)onM×[0,T]with uniformly bounded curvature. The Ricci ﬂow is a heat type equation. It is well-known that th e uniqueness of a heat equation on a complete noncompact manifold is not alway s held if there are no further restrictions on the growth of the solutions. For exa mple, the heat equation on Euclidean space with zero initial data has a nontrivial so lution which grows faster than exp(a|x|2) for anya>0 whenever t>0. This implies that even for the standard linear heat equation on Euclidean space, in order to ensure t he uniqueness one can only allow the solution to grow at most as exp( C|x|2) for some constant C >0. Note that on a K¨ ahler manifold, the Ricci curvature is given by Rα¯β=−∂2 ∂zα∂¯zβlog det(gγ¯δ). So the reasonable growth rate for the uniqueness of the Ricci ﬂow to hold is that the solution has bounded curvature. Thus the following uniquen ess result of Bing-Long Chen and the second author [33] is essentially the best one ca n hope for. Theorem 1.2.4 ( Chen-Zhu [33] ).Let(M,ˆgij)be a complete noncompact Rie- mannian manifold of dimension nwith bounded curvature. Let gij(x,t)and¯gij(x,t) be two solutions, deﬁned on M×[0,T], to the Ricci ﬂow (1.1.5)withˆgijas initial data and with bounded curvatures. Then gij(x,t)≡¯gij(x,t)onM×[0,T]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 183 1.3. Evolution of Curvatures. The Ricci ﬂow is an evolution equation on the metric. The evolution for the metric implies a nonlinear heat equation for the Riemannian curvature tensor Rijklwhich we will now derive. Proposition 1.3.1 ( Hamilton [58] ).Under the Ricci ﬂow (1.1.5),the curvature tensor satisﬁes the evolution equation ∂ ∂tRijkl= ∆Rijkl+ 2(Bijkl−Bijlk−Biljk+Bikjl) −gpq(RpjklRqi+RipklRqj+RijplRqk+RijkpRql) whereBijkl=gprgqsRpiqjRrksland∆is the Laplacian with respect to the evolving metric. Proof. Choose {x1,...,xm}to be a normal coordinate system at a ﬁxed point. At this point, we compute ∂ ∂tΓh jl=1 2ghm/braceleftbigg∂ ∂xj/parenleftbigg∂ ∂tglm/parenrightbigg +∂ ∂xl/parenleftbigg∂ ∂tgjm/parenrightbigg −∂ ∂xm/parenleftbigg∂ ∂tgjl/parenrightbigg/bracerightbigg =1 2ghm(∇j(−2Rlm) +∇l(−2Rjm)− ∇ m(−2Rjl)), ∂ ∂tRh ijl=∂ ∂xi/parenleftbigg∂ ∂tΓh jl/parenrightbigg −∂ ∂xj/parenleftbigg∂ ∂tΓh il/parenrightbigg , ∂ ∂tRijkl=ghk∂ ∂tRh ijl+∂ghk ∂tRh ijl. Combining these identities we get ∂ ∂tRijkl=ghk/bracketleftbigg/parenleftbigg1 2∇i[ghm(∇j(−2Rlm) +∇l(−2Rjm)− ∇ m(−2Rjl))]/parenrightbigg −/parenleftbigg1 2∇j[ghm(∇i(−2Rlm) +∇l(−2Rim)− ∇ m(−2Ril))]/parenrightbigg/bracketrightbigg −2RhkRh ijl =∇i∇kRjl− ∇ i∇lRjk− ∇ j∇kRil+∇j∇lRik −RijlpgpqRqk−RijkpgpqRql−2RijplgpqRqk =∇i∇kRjl− ∇ i∇lRjk− ∇ j∇kRil+∇j∇lRik −gpq(RijkpRql+RijplRqk). Here we have used the exchanging formula (1.1.3). Now it remains to check the following identity, which is anal ogous to the Simon′s identity in extrinsic geometry, ∆Rijkl+ 2(Bijkl−Bijlk−Biljk+Bikjl) (1.3.1) =∇i∇kRjl− ∇ i∇lRjk− ∇ j∇kRil+∇j∇lRik +gpq(RpjklRqi+RipklRqj). Indeed, from the second Bianchi identity (1.1.4), we have ∆Rijkl=gpq∇p∇qRijkl =gpq∇p∇iRqjkl−gpq∇p∇jRqikl. 184 H.-D. CAO AND X.-P. ZHU Let us examine the ﬁrst term on the RHS. By using the exchangin g formula (1.1.3) and the ﬁrst Bianchi identity (1.1.1), we have gpq∇p∇iRqjkl−gpq∇i∇pRqjkl =gpqgmn(RpiqmRnjkl+RpijmRqnkl+RpikmRqjnl+RpilmRqjkn) =RimgmnRnjkl+gpqgmnRpimj(Rqkln+Rqlnk) +gpqgmnRpikmRqjnl+gpqgmnRpilmRqjkn =RimgmnRnjkl−Bijkl+Bijlk−Bikjl+Biljk, while using the contracted second Bianchi identity (1.3.2) gpq∇pRqjkl=∇kRjl− ∇ lRjk, we have gpq∇i∇pRqjkl=∇i∇kRjl− ∇ i∇lRjk. Thus gpq∇p∇iRqjkl =∇i∇kRjl− ∇ i∇lRjk−(Bijkl−Bijlk−Biljk+Bikjl) +gpqRpjklRqi. Therefore we obtain ∆Rijkl =gpq∇p∇iRqjkl−gpq∇p∇jRqikl =∇i∇kRjl− ∇ i∇lRjk−(Bijkl−Bijlk−Biljk+Bikjl) +gpqRpjklRqi − ∇ j∇kRil+∇j∇lRik+ (Bjikl−Bjilk−Bjlik+Bjkil)−gpqRpiklRqj =∇i∇kRjl− ∇ i∇lRjk− ∇ j∇kRil+∇j∇lRik +gpq(RpjklRqi+RipklRqj)−2(Bijkl−Bijlk−Biljk+Bikjl) as desired, where in the last step we used the symmetries (1.3.3) Bijkl=Bklij=Bjilk. Corollary 1.3.2. The Ricci curvature satisﬁes the evolution equation ∂ ∂tRik= ∆Rik+ 2gprgqsRpiqkRrs−2gpqRpiRqk. Proof. ∂ ∂tRik=gjl∂ ∂tRijkl+/parenleftbigg∂ ∂tgjl/parenrightbigg Rijkl =gjl[∆Rijkl+ 2(Bijkl−Bijlk−Biljk+Bikjl) −gpq(RpjklRqi+RipklRqj+RijplRqk+RijkpRql)] −gjp/parenleftbigg∂ ∂tgpq/parenrightbigg gqlRijkl = ∆Rik+ 2gjl(Bijkl−2Bijlk) + 2gprgqsRpiqkRrs −2gpqRpkRqi. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 185 We claim that gjl(Bijkl−2Bijlk) = 0. Indeed by using the ﬁrst Bianchi identity, we have gjlBijkl=gjlgprgqsRpiqjRrksl =gjlgprgqsRpqijRrskl =gjlgprgqs(Rpiqj−Rpjqi)(Rrksl−Rrlsk) = 2gjl(Bijkl−Bijlk) as desired. Thus we obtain ∂ ∂tRik= ∆Rik+ 2gprgqsRpiqkRrs−2gpqRpiRqk. Corollary 1.3.3. The scalar curvature satisﬁes the evolution equation ∂R ∂t= ∆R+ 2|Ric|2. Proof. ∂R ∂t=gik∂Rik ∂t+/parenleftbigg −gip∂gpq ∂tgqk/parenrightbigg Rik =gik(∆Rik+ 2gprgqsRpiqkRrs−2gpqRpiRqk) + 2RpqRikgipgqk = ∆R+ 2|Ric|2. To simplify the evolution equations of curvatures, we will r epresent the curvature tensors in an orthonormal frame and evolve the frame so that i t remains orthonor- mal. More precisely, let us pick an abstract vector bundle VoverMisomorphic to the tangent bundle TM. Locally, the frame F={F1,...,F a,...,F n}ofVis given byFa=Fi a∂ ∂xiwith the isomorphism {Fi a}. Choose {Fi a}att= 0 such that F={F1,...,F a,...,F n}is an orthonormal frame at t= 0, and evolve {Fi a}by the equation ∂ ∂tFi a=gijRjkFk a. Then the frame F={F1,...,F a,...,F n}will remain orthonormal for all times since the pull back metric on V hab=gijFi aFj b remains constant in time. In the following we will use indice sa,b,... on a tensor to denote its components in the evolving orthonormal frame. In this frame we have the 186 H.-D. CAO AND X.-P. ZHU following: Rabcd=Fi aFj bFk cFl dRijkl, Γa jb=Fa i∂Fi b ∂xj+ Γi jkFa iFk b,((Fa i) = (Fi a)−1) ∇iVa=∂ ∂xiVa+ Γa ibVb, ∇bVa=Fi b∇iVa, where Γa jbis the metric connection of the vector bundle Vwith the metric hab. Indeed, by direct computations, ∇iFj b=∂Fj b ∂xi+Fk bΓj ik−Fj cΓc ib =∂Fj b ∂xi+Fk bΓj ik−Fj c/parenleftbigg Fc k∂Fk b ∂xi+ Γl ikFc lFk b/parenrightbigg = 0, ∇ihab=∇i(gijFi aFj b) = 0. So ∇aVb=Fi aFj b∇iVj, and ∆Rabcd=∇l∇lRabcd =gij∇i∇jRabcd =gijFk aFl bFm cFn d∇i∇jRklmn. In an orthonormal frame F={F1,...,F a,...,F n}, the evolution equations of curvature tensors become ∂ ∂tRabcd= ∆Rabcd+ 2(Babcd−Babdc−Badbc+Bacbd) (1.3.4) ∂ ∂tRab= ∆Rab+ 2RacbdRcd (1.3.5) ∂ ∂tR= ∆R+ 2|Ric|2(1.3.6) whereBabcd=RaebfRced f. Equation (1.3.4) is a reaction-diﬀusion equation. We can un derstand the quadratic terms of this equation better if we think of the cur vature tensor Rabcd as a symmetric bilinear form on the two-forms Λ2(V) given by the formula Rm(ϕ,ψ) =Rabcdϕabψcd,forϕ,ψ∈Λ2(V). A two-form ϕ∈Λ2(V) can be regarded as an element of the Lie algebra so(n) (i.e. the skew-symmetric matrix ( ϕab)n×n), where the metric on Λ2(V) is given by /a\}b∇acketle{tϕ,ψ/a\}b∇acket∇i}ht=ϕabψab THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 187 and the Lie bracket is given by [ϕ,ψ]ab=ϕacψbc−ψacϕbc. Choose an orthonormal basis of Λ2(V) Φ ={ϕ1,...,ϕα,...,ϕn(n−1) 2} whereϕα={ϕα ab}. The Lie bracket is given by [ϕα,ϕβ] =Cαβ γϕγ, whereCαβγ=Cαβ σδσγ=/a\}b∇acketle{t[ϕα,ϕβ],ϕγ/a\}b∇acket∇i}htare the Lie structure constants. WriteRabcd=Mαβϕα abϕβ cd.We now claim that the ﬁrst part of the quadratic terms in (1.3.4) is given by (1.3.7) 2( Babcd−Babdc) =MαγMβγϕα abϕβ cd. Indeed, by the ﬁrst Bianchi identity, Babcd−Babdc=RaebfRced f−RaebfRdecf =Raebf(−Rcefd−Rcfde) =RaebfRcdef. On the other hand, RaebfRcdef= (−Rabfe−Rafeb)Rcdef =RabefRcdef−RafebRcdef =RabefRcdef−RafbeRcd fe which implies RaebfRcdef=1 2RabefRcdef. Thus we obtain 2(Babcd−Babdc) =RabefRcdef=MαγMβγϕα abϕβ cd. We next consider the last part of the quadratic terms: 2(Bacbd−Badbc) = 2(RaecfRbed f−Raed fRbecf) = 2(Mγδϕγ aeϕδ cfMηθϕη beϕθ d f−Mγθϕγ aeϕθ d fMηδϕη beϕδ cf) = 2[Mγδ(ϕη aeϕγ be+Cγη αϕα ab)ϕδ cfMηθϕθ d f−Mηθϕη aeϕθ d fMγδϕγ beϕδ cf] = 2Mγδϕδ cfMηθϕθ d fCγη αϕα ab. But Mγδϕδ cfMηθϕθ d fCγη αϕα ab =MγδMηθCγη αϕα ab[ϕθ cfϕδ d f+Cδθ βϕβ cd] =−MηθMγδCγη αϕα abϕδ cfϕθ d f+MγδMηθCγη αCδθ βϕα abϕβ cd =−MηθMγδCγη αϕα abϕδ cfϕθ d f+ (Cγη αCδθ βMγδMηθ)ϕα abϕβ cd 188 H.-D. CAO AND X.-P. ZHU which implies Mγδϕδ cfMηθϕθ d fCγη αϕα ab=1 2(Cγη αCδθ βMγδMηθ)ϕα abϕβ cd. Then we have (1.3.8) 2( Bacbd−Badbc) = (Cγη αCδθ βMγδMηθ)ϕα abϕβ cd. Therefore, combining (1.3.7) and (1.3.8), we can reformula te the curvature evo- lution equation (1.3.4) as follows. Proposition 1.3.4 ( Hamilton [59] ).LetRabcd=Mαβϕα abϕβ cd.Then under the Ricci ﬂow (1.1.5), Mαβsatisﬁes the evolution equation (1.3.9)∂Mαβ ∂t= ∆Mαβ+M2 αβ+M# αβ whereM2 αβ=MαγMβγis the operator square and M# αβ= (Cγη αCδθ βMγδMηθ)is the Lie algebra square. Let us now consider the operator M# αβin dimensions 3 and 4 in more detail. In dimension 3, let ω1,ω2,ω3be a positively oriented orthonormal basis for one- forms. Then ϕ1=√ 2ω1∧ω2, ϕ2=√ 2ω2∧ω3, ϕ3=√ 2ω3∧ω1 form an orthonormal basis for two-forms Λ2. Writeϕα={ϕα ab},α= 1,2,3,as (ϕ1 ab) = 0√ 2 20 −√ 2 20 0 0 0 0 ,(ϕ2 ab) = 0 0 0 0 0√ 2 2 0−√ 2 20 , (ϕ3 ab) = 0 0 −√ 2 2 0 0 0√ 2 20 0 , then [ϕ1,ϕ2] = 0√ 2 20 −√ 2 20 0 0 0 0  0 0 0 0 0−√ 2 2 0√ 2 20 − 0 0 0 0 0√ 2 2 0−√ 2 20  0−√ 2 20√ 2 20 0 0 0 0  = 0 0−1 2 0 0 0 1 20 0  =√ 2 2ϕ3. SoC123=/a\}b∇acketle{t[ϕ1,ϕ2],ϕ3/a\}b∇acket∇i}ht=√ 2 2, in particular Cαβγ=/braceleftigg ±√ 2 2,ifα/\e}atio\slash=β/\e}atio\slash=γ, 0, otherwise. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 189 Hence the matrix M#= (M# αβ) is just the adjoint matrix of M= (Mαβ): (1.3.10) M#= detM·tM−1. In dimension 4, we can use the Hodge star operator to decompos e the space of two-forms Λ2as Λ2= Λ2 +⊕Λ2 − where Λ2 +(resp. Λ2 −) is the eigenspace of the star operator with eigenvalue +1 (r esp. −1). Letω1,ω2,ω3,ω4be a positively oriented orthonormal basis for one-forms. A basis for Λ2 +is then given by ϕ1=ω1∧ω2+ω3∧ω4, ϕ2=ω1∧ω3+ω4∧ω2, ϕ3=ω1∧ω4+ω2∧ω3, while a basis for Λ2 −is given by ψ1=ω1∧ω2−ω3∧ω4, ψ2=ω1∧ω3−ω4∧ω2, ψ3=ω1∧ω4−ω2∧ω3. In particular, {ϕ1,ϕ2,ϕ3,ψ1,ψ2,ψ3}forms an orthonormal basis for the space of two-forms Λ2. By using this basis we obtain a block decomposition of the cu rvature operator matrix Mas M= (Mαβ) =/parenleftbigg A B tB C/parenrightbigg . HereA,BandCare 3×3 matrices with AandCbeing symmetric. Then we can write each element of the basis as a skew-symmetric 4 ×4 matrix and compute as above to get (1.3.11) M#= (M# αβ) = 2/parenleftbiggA#B# tB#C#/parenrightbigg , whereA#,B#,C#are the adjoint of 3 ×3 submatrices as before. For later applications in Chapter 5, we now give some computa tions for the entries of the matrices A,CandBas follows. First for the matrices AandC, we have A11=Rm(ϕ1,ϕ1) =R1212+R3434+ 2R1234 A22=Rm(ϕ2,ϕ2) =R1313+R4242+ 2R1342 A33=Rm(ϕ3,ϕ3) =R1414+R2323+ 2R1423 and C11=Rm(ψ1,ψ1) =R1212+R3434−2R1234 C22=Rm(ψ2,ψ2) =R1313+R4242−2R1342 C33=Rm(ψ3,ψ3) =R1414+R2323−2R1423. 190 H.-D. CAO AND X.-P. ZHU By the Bianchi identity R1234+R1342+R1423= 0, so we have trA=trC=1 2R. Next for the entries of the matrix B, we have B11=Rm(ϕ1,ψ1) =R1212−R3434 B22=Rm(ϕ2,ψ2) =R1313−R4242 B33=Rm(ϕ3,ψ3) =R1414−R2323 and B12=Rm(ϕ1,ψ2) =R1213+R3413−R1242−R3442 etc. Thus the entries of Bcan be written as B11=1 2(R11+R22−R33−R44) B22=1 2(R11+R33−R44−R22) B33=1 2(R11+R44−R22−R33) and B12=R23−R14etc. If we choose the frame {ω1,ω2,ω3,ω4}so that the Ricci tensor is diagonal, then the matrixBis also diagonal. In particular, the matrix Bis identically zero when the four-manifold is Einstein. 1.4. Derivative Estimates. In the previous section we have seen that the cur- vatures satisfy nonlinear heat equations with quadratic gr owth terms. The parabolic nature will give us a bound on the derivatives of the curvatur es at any time t>0 in terms of a bound of the curvatures. We begin with the global version of the derivative estimates . Theorem 1.4.1 ( Shi [114] ).There exist constants Cm,m= 1,2,...,such that if the curvature of a complete solution to Ricci ﬂow is bounde d by |Rijkl| ≤M up to time t with 0<t≤1 M, then the covariant derivative of the curvature is bounded by |∇Rijkl| ≤C1M/√ t THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 191 and themthcovariant derivative of the curvature is bounded by |∇mRijkl| ≤CmM/tm 2. Here the norms are taken with respect to the evolving metric. Proof. We shall only give the proof for the compact case. The noncomp act case can be deduced from the next local derivative estimate theor em. Let us denote the curvature tensor by Rmand denote by A∗Bany tensor product of two tensors A andBwhen we do not need the precise expression. We have from Propo sition 1.3.1 that (1.4.1)∂ ∂tRm= ∆Rm+Rm∗Rm. Since ∂ ∂tΓi jk=1 2gil/braceleftbigg ∇j∂gkl ∂t+∇k∂gjl ∂t− ∇ l∂gjk ∂t/bracerightbigg =∇Rm, it follows that (1.4.2)∂ ∂t(∇Rm) = ∆( ∇Rm) +Rm∗(∇Rm). Thus ∂ ∂t|Rm|2≤∆|Rm|2−2|∇Rm|2+C|Rm|3, ∂ ∂t|∇Rm|2≤∆|∇Rm|2−2|∇2Rm|2+C|Rm| · |∇Rm|2, for some constant Cdepending only on the dimension n. LetA>0 be a constant (to be determined) and set F=t|∇Rm|2+A|Rm|2. We compute ∂F ∂t=|∇Rm|2+t∂ ∂t|∇Rm|2+A∂ ∂t|Rm|2 ≤∆(t|∇Rm|2+A|Rm|2) +|∇Rm|2(1 +tC|Rm| −2A) +CA|Rm|3. TakingA≥C+ 1,we get ∂F ∂t≤∆F+¯CM3 for some constant ¯Cdepending only on the dimension n. We then obtain F≤F(0) + ¯CM3t≤(A+¯C)M2, and then |∇Rm|2≤(A+¯C)M2/t. 192 H.-D. CAO AND X.-P. ZHU The general case follows in the same way. If we have bounds |∇kRm| ≤CkM/tk 2, we know from (1.4.1) and (1.4.2) that ∂ ∂t|∇kRm|2≤∆|∇kRm|2−2|∇k+1Rm|2+CM3 tk, and ∂ ∂t|∇k+1Rm|2≤∆|∇k+1Rm|2−2|∇k+2Rm|2+CM|∇k+1Rm|2+CM3 tk+1. LetAk>0 be a constant (to be determined) and set Fk=tk+2|∇k+1Rm|2+Aktk+1|∇kRm|2. Then ∂ ∂tFk= (k+ 2)tk+1|∇k+1Rm|2+tk+2∂ ∂t|∇k+1Rm|2 +Ak(k+ 1)tk|∇kRm|2+Aktk+1∂ ∂t|∇kRm|2 ≤(k+ 2)tk+1|∇k+1Rm|2 +tk+2/bracketleftbigg ∆|∇k+1Rm|2−2|∇k+2Rm|2+CM|∇k+1Rm|2+CM3 tk+1/bracketrightbigg +Ak(k+ 1)tk|∇kRm|2 +Aktk+1/bracketleftbigg ∆|∇kRm|2−2|∇k+1Rm|2+CM3 tk/bracketrightbigg ≤∆Fk+Ck+1M2 for some positive constant Ck+1, by choosing Aklarge enough. This implies that |∇k+1Rm| ≤Ck+1M tk+1 2. The above derivative estimate is a somewhat standard Bernst ein estimate in PDEs. By using a cutoﬀ argument, we will derive the following local version, which is called Shi’s derivative estimate . The following proof is adapted from Hamilton [63]. Theorem 1.4.2 ( Shi [114] ).There exist positive constants θ,Ck,k= 1,2,...,de- pending only on the dimension with the following property. S uppose that the curvature of a solution to the Ricci ﬂow is bounded |Rm| ≤M, onU×/bracketleftbigg 0,θ M/bracketrightbigg whereUis an open set of the manifold. Assume that the closed ball B0(p,r), centered atpof radiusrwith respect to the metric at t= 0, is contained in Uand the time t≤θ/M. Then we can estimate the covariant derivatives of the curva ture at (p,t)by |∇Rm(p,t)|2≤C1M2/parenleftbigg1 r2+1 t+M/parenrightbigg , THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 193 and thekthcovariant derivative of the curvature at (p,t)by |∇kRm(p,t)|2≤CkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg . Proof. Without loss of generality, we may assume r≤θ/√ Mand the exponential map atpat timet= 0 is injective on the ball of radius r(by passing to a local cover if necessary, and pulling back the local solution of the Ricc i ﬂow to the ball of radius rin the tangent space at pat timet= 0). Recall ∂ ∂t|Rm|2≤∆|Rm|2−2|∇Rm|2+C|Rm|3, ∂ ∂t|∇Rm|2≤∆|∇Rm|2−2|∇2Rm|2+C|Rm| · |∇Rm|2. Deﬁne S= (BM2+|Rm|2)|∇Rm|2 whereBis a positive constant to be determined. By choosing B≥C2/4 and using the Cauchy inequality, we have ∂ ∂tS≤∆S−2BM2|∇2Rm|2−2|∇Rm|4 +CM|∇Rm|2· |∇2Rm|+CBM3|∇Rm|2 ≤∆S− |∇Rm|4+CB2M6 ≤∆S−S2 (B+ 1)2M4+CB2M6. If we take F=b(BM2+|Rm|2)|∇Rm|2/M4=bS/M4, andb≤min{1/(B+ 1)2,1/CB2}, we get (1.4.3)∂F ∂t≤∆F−F2+M2. We now want to choose a cutoﬀ function ϕwith the support in the ball B0(p,r) such that at t= 0, ϕ(p) =r,0≤ϕ≤Ar, and |∇ϕ| ≤A,|∇2ϕ| ≤A r for some positive constant Adepending only on the dimension. Indeed, let g: (−∞,+∞)→[0,+∞) be a smooth, nonnegative function satisfying g(u) =/braceleftigg 1, u∈(−1 2,1 2), 0,outside ( −1,1). 194 H.-D. CAO AND X.-P. ZHU Set ϕ=rg/parenleftbiggs2 r2/parenrightbigg , wheresis the geodesic distance function from pwith respect to the metric at t= 0. Then ∇ϕ=1 rg′/parenleftbiggs2 r2/parenrightbigg ·2s∇s and hence |∇ϕ| ≤2C1. Also, ∇2ϕ=1 rg′′/parenleftbiggs2 r2/parenrightbigg1 r24s2∇s· ∇s+1 rg′/parenleftbiggs2 r2/parenrightbigg 2∇s· ∇s+1 rg′/parenleftbiggs2 r2/parenrightbigg ·2s∇2s. Thus, by using the standard Hessian comparison, |∇2ϕ| ≤C1 r+C1 r|s∇2s| ≤C1 r/parenleftbigg 1 +s/parenleftbiggC2 s+√ M/parenrightbigg/parenrightbigg ≤C3 r. HereC1,C2andC3are positive constants depending only on the dimension. Now extend ϕtoU×[0,θ M] by letting ϕto be zero outside B0(p,r) and indepen- dent of time. Introduce the barrier function (1.4.4) H=(12 + 4√n)A2 ϕ2+1 t+M which is deﬁned and smooth on the set {ϕ>0} ×(0,T]. As the metric evolves, we will still have 0 ≤ϕ≤Ar(sinceϕis independent of timet); but |∇ϕ|2andϕ|∇2ϕ|may increase. By continuity it will be a while before they double. Claim 1. As long as |∇ϕ|2≤2A2, ϕ|∇2ϕ| ≤2A2, we have ∂H ∂t>∆H−H2+M2. Indeed, by the deﬁnition of H, we have H2>(12 + 4√n)2A4 ϕ4+1 t2+M2, ∂H ∂t=−1 t2, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 195 and ∆H= (12 + 4√n)A2∆/parenleftbigg1 ϕ2/parenrightbigg = (12 + 4√n)A2/parenleftbigg6|∇ϕ|2−2ϕ∆ϕ ϕ4/parenrightbigg ≤(12 + 4√n)A2/parenleftbigg12A2+ 4√nA2 ϕ4/parenrightbigg =(12 + 4√n)2A4 ϕ4. Therefore, H2>∆H−∂H ∂t+M2. Claim 2. If the constant θ >0 is small enough compared to b,BandA, then we have the following property: as long as r≤θ/√ M,t≤θ/MandF≤H, we will have |∇ϕ|2≤2A2andϕ|∇2ϕ| ≤2A2. Indeed, by considering the evolution of ∇ϕ, we have ∂ ∂t∇aϕ=∂ ∂t(Fi a∇iϕ) =Fi a∇i/parenleftbigg∂ϕ ∂t/parenrightbigg +∇iϕRi kFk a =Rab∇bϕ which implies ∂ ∂t|∇ϕ|2≤CM|∇ϕ|2, and then |∇ϕ|2≤A2eCMt≤2A2, providedt≤θ/Mwithθ≤log 2/C. By considering the evolution of ∇2ϕ, we have ∂ ∂t(∇a∇bϕ) =∂ ∂t(Fi aFj b∇i∇jϕ) =∂ ∂t/parenleftbigg Fi aFj b/parenleftbigg∂2ϕ ∂xi∂xj−Γk ij∂ϕ ∂xk/parenrightbigg/parenrightbigg =∇a∇b/parenleftbigg∂ϕ ∂t/parenrightbigg +Rac∇b∇cϕ+Rbc∇a∇cϕ + (∇cRab− ∇ aRbc− ∇ bRac)∇cϕ 196 H.-D. CAO AND X.-P. ZHU which implies (1.4.5)∂ ∂t|∇2ϕ| ≤C|Rm| · |∇2ϕ|+C|∇Rm| · |∇ϕ|. By assumption F≤H, we have (1.4.6) |∇Rm|2≤2M2 bB/parenleftbigg(12 + 4√n)A2 ϕ2+1 t/parenrightbigg ,fort≤θ/M. Thus by noting ϕindependent of tandϕ≤Ar, we get from (1.4.5) and (1.4.6) that ∂ ∂t(ϕ|∇2ϕ|)≤CM/parenleftbigg ϕ|∇2ϕ|+ 1 +r√ t/parenrightbigg which implies ϕ|∇2ϕ| ≤eCMt/bracketleftbigg (ϕ|∇2ϕ|)|t=0+CM/integraldisplayt 0/parenleftbigg 1 +r√ t/parenrightbigg dt/bracketrightbigg ≤eCMt/bracketleftig A2+CM(t+ 2r√ t)/bracketrightig ≤2A2 providedr≤θ/√ M, andt≤θ/Mwithθsmall enough. Therefore we have obtained Claim 2. The combination of Claim 1 and Claim 2 gives us ∂H ∂t>∆H−H2+M2 as long asr≤θ/√ M, t ≤θ/MandF≤H. And (1.4.3) tells us ∂F ∂t≤∆F−F2+M2. Then the standard maximum principle immediately gives the e stimate |∇Rm|2≤CM2/parenleftbigg1 ϕ2+1 t+M/parenrightbigg on{ϕ>0} ×/parenleftig 0,θ M/bracketrightig , which implies the ﬁrst order derivative estimate. The higher order derivative estimates can be obtained in the same way by induc- tion. Suppose we have the bounds |∇kRm|2≤CkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg . As before, by (1.4.1) and (1.4.2), we have ∂ ∂t|∇kRm|2≤∆|∇kRm|2−2|∇k+1Rm|2+CM3/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg , and ∂ ∂t|∇k+1Rm|2≤∆|∇k+1Rm|2−2|∇k+2Rm|2 +CM|∇k+1Rm|2+CM3/parenleftbigg1 r2(k+1)+1 tk+1+Mk+1/parenrightbigg . THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 197 Here and in the following we denote by Cvarious positive constants depending only onCkand the dimension. Deﬁne Sk=/bracketleftbigg BkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg +|∇kRm|2/bracketrightbigg · |∇k+1Rm|2 whereBkis a positive constant to be determined. By choosing Bklarge enough and Cauchy inequality, we have ∂ ∂tSk≤/bracketleftbigg −k tk+1+ ∆|∇kRm|2−2|∇k+1Rm|2 +CM3/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg/bracketrightbigg · |∇k+1Rm|2 +/bracketleftbigg BkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg +|∇kRm|2/bracketrightbigg ·/bracketleftbigg ∆|∇k+1Rm|2−2|∇k+2Rm|2+CM|∇k+1Rm|2 +CM3/parenleftbigg1 r2(k+1)+1 tk+1+Mk+1/parenrightbigg/bracketrightbigg ≤∆Sk+ 8|∇kRm| · |∇k+1Rm|2· |∇k+2Rm| −k tk+1|∇k+1Rm|2 −2|∇k+1Rm|4+CM3|∇k+1Rm|2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg −2|∇k+2Rm|2/bracketleftbigg BkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg +|∇kRm|2/bracketrightbigg +CM|∇k+1Rm|2/bracketleftbigg BkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg +|∇kRm|2/bracketrightbigg +CM3/parenleftbigg1 r2(k+1)+1 tk+1+Mk+1/parenrightbigg ·/bracketleftbigg BkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg +|∇kRm|2/bracketrightbigg ≤∆Sk− |∇k+1Rm|4+CB2 kM6/parenleftbigg1 r4k+1 t2k+M2k/parenrightbigg +CBkM5/parenleftbigg1 r2(2k+1)+1 t2k+1+M2k+1/parenrightbigg ≤∆Sk− |∇k+1Rm|4+CB2 kM5/parenleftbigg1 r2(2k+1)+1 t2k+1+M2k+1/parenrightbigg ≤∆Sk−Sk (B+ 1)2M4/parenleftbig1 r2k+1 tk+Mk/parenrightbig2 +CB2 kM5/parenleftbigg1 r2(2k+1)+1 t2k+1+M2k+1/parenrightbigg . 198 H.-D. CAO AND X.-P. ZHU Letu= 1/r2+ 1/t+Mand setFk=bSk/uk. Then ∂Fk ∂t≤∆Fk−F2 k b(Bk+ 1)2M4uk+bCB2 kM5uk+1+kFku ≤∆Fk−F2 k 2b(Bk+ 1)2M4uk+b(C+ 2k2)(Bk+ 1)2M4uk+2. By choosing b≤1/(2(C+ 2k2)(Bk+ 1)2M4), we get ∂Fk ∂t≤∆Fk−1 ukF2 k+uk+2. Introduce Hk= 5(k+ 1)(2(k+ 1) + 1 +√n)A2ϕ−2(k+1)+Lt−(k+1)+Mk+1, whereL≥k+ 2. Then by using Claim 1 and Claim 2, we have ∂Hk ∂t=−(k+ 1)Lt−(k+2), ∆Hk≤20(k+ 1)2(2(k+ 1) + 1 +√n)A4ϕ−2(k+2) and H2 k>25(k+ 1)2(2(k+ 1) + 1 +√n)A4ϕ−4(k+1)+L2t−2(k+1)+M2(k+1). These imply ∂Hk ∂t>∆Hk−1 ukH2 k+uk+2. Then the maximum principle immediately gives the estimate Fk≤Hk. In particular, b ukBkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg · |∇k+1Rm|2 ≤5(k+ 1)/parenleftbig 2(k+ 1) + 1 +√n/parenrightbig A2ϕ−2(k+1)+Lt−(k+1)+Mk+1. So by the deﬁnition of uand the choosing of b, we obtain the desired estimate |∇k+1Rm|2≤Ck+1M2/parenleftbigg1 r2(k+1)+1 tk+1+Mk+1/parenrightbigg . Therefore we have completed the proof of the theorem. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 199 1.5. Variational Structure and Dynamic Property. In this section, we in- troduce two functionals of Perelman [103], FandW, and discuss their relations with the Ricci ﬂow. It was not known whether the Ricci ﬂow is a gradi ent ﬂow until Perel- man [103] showed that the Ricci ﬂow is, in a certain sense, the gradient ﬂow of the functional F. If we consider the Ricci ﬂow as a dynamical system on the spac e of Rie- mannian metrics, then these two functionals are of Lyapunov type for this dynamical system. Obviously, Ricci ﬂat metrics are ﬁxed points of the d ynamical system. When we consider the space of Riemannian metrics modulo diﬀeomor phism and scaling, ﬁxed points of the Ricci ﬂow dynamical system correspond to s teady, or shrinking, or expanding Ricci solitons. The following concept correspon ds to a periodic orbit. Definition 1.5.1. A metricgij(t) evolving by the Ricci ﬂow is called a breather if forsomet1< t2andα >0 the metrics αgij(t1) andgij(t2) diﬀer only by a diﬀeomorphism; the case α= 1, α< 1, α> 1 correspond to steady, shrinking and expanding breathers , respectively. Clearly, (steady, shrinking or expanding) Ricci solitons a re trivial breathers for which the metrics gij(t1) andgij(t2) diﬀer only by diﬀeomorphism and scaling for every pairt1andt2. We always assume Mis a compact n-dimensional manifold in this section. Let us ﬁrst consider the functional (1.5.1) F(gij,f) =/integraldisplay M(R+|∇f|2)e−fdV of Perelman [103] deﬁned on the space of Riemannian metrics, and smooth functions onM. HereRis the scalar curvature of gij. Lemma 1.5.2 ( Perelman [103] ).Ifδgij=vijandδf=hare variations of gij andfrespectively, then the ﬁrst variation of Fis given by δF(vij,h) =/integraldisplay M/bracketleftig −vij(Rij+∇i∇jf) +/parenleftigv 2−h/parenrightig (2∆f− |∇f|2+R)/bracketrightig e−fdV wherev=gijvij. 200 H.-D. CAO AND X.-P. ZHU Proof. In any normal coordinates at a ﬁxed point, we have δRh ijl=∂ ∂xi(δΓh jl)−∂ ∂xj(δΓh il) =∂ ∂xi/bracketleftbigg1 2ghm(∇jvlm+∇lvjm− ∇ mvjl)/bracketrightbigg −∂ ∂xj/bracketleftbigg1 2ghm(∇ivlm+∇lvim− ∇ mvil)/bracketrightbigg , δRjl=∂ ∂xi/bracketleftbigg1 2gim(∇jvlm+∇lvjm− ∇ mvjl)/bracketrightbigg −∂ ∂xj/bracketleftbigg1 2gim(∇ivlm+∇lvim− ∇ mvil)/bracketrightbigg =1 2∂ ∂xi[∇jvi l+∇lvi j− ∇ivjl]−1 2∂ ∂xj[∇lv], δR=δ(gjlRjl) =−vjlRjl+gjlδRjl =−vjlRjl+1 2∂ ∂xi[∇lvi l+∇lvil− ∇iv]−1 2∂ ∂xj[∇jv] =−vjlRjl+∇i∇lvil−∆v. Thus (1.5.2) δR(vij) =−∆v+∇i∇jvij−vijRij. The ﬁrst variation of the functional F(gij,f) is δ/parenleftbigg/integraldisplay M(R+|∇f|2)e−fdV/parenrightbigg (1.5.3) =/integraldisplay M/parenleftbigg [δR(vij) +δ(gij∇if∇jf)]e−fdV + (R+|∇f|2)/bracketleftig −he−fdV+e−fv 2dV/bracketrightig/parenrightbigg =/integraldisplay M/bracketleftbigg −∆v+∇i∇jvij−Rijvij−vij∇if∇jf + 2/a\}b∇acketle{t∇f,∇h/a\}b∇acket∇i}ht+ (R+|∇f|2)/parenleftigv 2−h/parenrightig/bracketrightbigg e−fdV. On the other hand, /integraldisplay M(∇i∇jvij−vij∇if∇jf)e−fdV=/integraldisplay M(∇if∇jvij−vij∇if∇jf)e−fdV =−/integraldisplay M(∇i∇jf)vije−fdV, /integraldisplay M2/a\}b∇acketle{t∇f,∇h/a\}b∇acket∇i}hte−fdV=−2/integraldisplay Mh∆fe−fdV+ 2/integraldisplay M|∇f|2he−fdV, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 201 and /integraldisplay M(−∆v)e−fdV=−/integraldisplay M/a\}b∇acketle{t∇f,∇v/a\}b∇acket∇i}hte−fdV =/integraldisplay Mv∆fe−fdV−/integraldisplay M|∇f|2ve−fdV. Plugging these identities into (1.5.3) the ﬁrst variation f ormula follows. Now let us study the functional Fwhen the metric evolves under the Ricci ﬂow and the function evolves by a backward heat equation. Proposition 1.5.3 ( Perelman [103] ).Letgij(t)andf(t)evolve according to the coupled ﬂow /braceleftigg∂gij ∂t=−2Rij, ∂f ∂t=−∆f+|∇f|2−R. Then d dtF(gij(t),f(t)) = 2/integraldisplay M|Rij+∇i∇jf|2e−fdV and/integraltext Me−fdVis constant. In particular F(gij(t),f(t))is nondecreasing in time and the monotonicity is strict unless we are on a steady gradient soliton. Proof. Under the coupled ﬂow and using the ﬁrst variation formula in Lemma 1.5.2, we have d dtF(gij(t),f(t)) =/integraldisplay M/bracketleftbigg −(−2Rij)(Rij+∇i∇jf) +/parenleftbigg1 2(−2R)−∂f ∂t/parenrightbigg (2∆f− |∇f|2+R)/bracketrightbigg e−fdV =/integraldisplay M[2Rij(Rij+∇i∇jf) + (∆f− |∇f|2)(2∆f− |∇f|2+R)]e−fdV. Now /integraldisplay M(∆f− |∇f|2)(2∆f− |∇f|2)e−fdV =/integraldisplay M−∇if∇i(2∆f− |∇f|2)e−fdV =/integraldisplay M−∇if(2∇j(∇i∇jf)−2Rij∇jf−2/a\}b∇acketle{t∇f,∇i∇f/a\}b∇acket∇i}ht)e−fdV =−2/integraldisplay M[(∇if∇jf−∇i∇jf)∇i∇jf−Rij∇if∇jf−/a\}b∇acketle{t∇f,∇i∇f/a\}b∇acket∇i}ht∇if]e−fdV = 2/integraldisplay M[|∇i∇jf|2+Rij∇if∇jf]e−fdV, 202 H.-D. CAO AND X.-P. ZHU and /integraldisplay M(∆f− |∇f|2)Re−fdV =/integraldisplay M−∇if∇iRe−fdV = 2/integraldisplay M∇i∇jfRije−fdV−2/integraldisplay M∇if∇jfRije−fdV. Here we have used the contracted second Bianchi identity. Th erefore we obtain d dtF(gij(t),f(t)) =/integraldisplay M[2Rij(Rij+∇i∇jf) + 2(∇i∇jf)(∇i∇jf+Rij)]e−fdV = 2/integraldisplay M|Rij+∇i∇jf|2e−fdV. It remains to show/integraltext Me−fdVis a constant. Note that the volume element dV=/radicalbig detgijdxevolves under the Ricci ﬂow by ∂ ∂tdV=∂ ∂t(/radicalbig detgij)dx (1.5.4) =1 2/parenleftbigg∂ ∂tlog(detgij)/parenrightbigg dV =1 2(gij∂ ∂tgij)dV =−RdV. Hence ∂ ∂t/parenleftbig e−fdV/parenrightbig =e−f/parenleftbigg −∂f ∂t−R/parenrightbigg dV (1.5.5) = (∆f− |∇f|2)e−fdV =−∆(e−f)dV. It then follows that d dt/integraldisplay Me−fdV=−/integraldisplay M∆(e−f)dV= 0. This ﬁnishes the proof of the proposition. Next we deﬁne the associated energy (1.5.6) λ(gij) = inf/braceleftbigg F(gij,f)|f∈C∞(M),/integraldisplay Me−fdV= 1/bracerightbigg . If we setu=e−f/2, then the functional Fcan be expressed in terms of uas F=/integraldisplay M(Ru2+ 4|∇u|2)dV, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 203 and the constraint/integraltext Me−fdV= 1 becomes/integraltext Mu2dV= 1. Therefore λ(gij) is just the ﬁrst eigenvalue of the operator −4∆ +R. Letu0>0 be a ﬁrst eigenfunction of the operator −4∆ +Rsatisfying −4∆u0+Ru0=λ(gij)u0. Thef0=−2 logu0is a minimizer: λ(gij) =F(gij,f0). Note thatf0satisﬁes the equation (1.5.7) −2∆f0+|∇f0|2−R=−λ(gij). Observe that the evolution equation ∂f ∂t=−∆f+|∇f|2−R can be rewritten as the following linear equation ∂ ∂t(e−f) =−∆(e−f) +R(e−f). Thus we can always solve the evolution equation for fbackwards in time. Suppose att=t0, the inﬁmum λ(gij) is achieved by some function f0with/integraltext Me−f0dV= 1. We solve the backward heat equation /braceleftigg∂f ∂t=−∆f+|∇f|2−R f|t=t0=f0 to obtain a solution f(t) fort≤t0which satisﬁes/integraltext Me−fdV= 1. It then follows from Proposition 1.5.3 that λ(gij(t))≤ F(gij(t),f(t))≤ F(gij(t0),f(t0)) =λ(gij(t0)). Also noteλ(gij) is invariant under diﬀeomorphism. Thus we have proved Corollary 1.5.4. (i)λ(gij(t))is nondecreasing along the Ricci ﬂow and the monotonicity is strict unless we are on a steady gradient soliton; (ii)A steady breather is necessarily a steady gradient soliton. To deal with the expanding case we consider a scale invariant version ¯λ(gij) =λ(gij)V2 n(gij). HereV=Vol(gij) denotes the volume of Mwith respect to the metric gij. 204 H.-D. CAO AND X.-P. ZHU Corollary 1.5.5. (i)¯λ(gij)is nondecreasing along the Ricci ﬂow whenever it is nonposit ive; more- over, the monotonicity is strict unless we are on a gradient e xpanding soliton; (ii)An expanding breather is necessarily an expanding gradient soliton. Proof. Letf0be a minimizer of λ(gij(t)) att=t0and solve the backward heat equation ∂f ∂t=−∆f+|∇f|2−R to obtainf(t),t≤t0, with/integraltext Me−f(t)dV= 1. We compute the derivative (understood in the barrier sense) at t=t0, d dt¯λ(gij(t)) ≥d dt(F(gij(t),f(t))·V2 n(gij(t))) =V2 n/integraldisplay M2|Rij+∇i∇jf|2e−fdV +2 nV2−n n/integraldisplay M(−R)dV·/integraldisplay M(R+|∇f|2)e−fdV = 2V2 n/bracketleftbigg/integraldisplay M/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 n(R+ ∆f)gij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 e−fdV +1 n/integraldisplay M(R+∆f)2e−fdV+1 n/parenleftbigg −/integraldisplay M(R+|∇f|2)e−fdV/parenrightbigg/parenleftbigg1 V/integraldisplay MRdV/parenrightbigg/bracketrightbigg , where we have used the formula (1.5.4) in the computation of dV/dt . Supposeλ(gij(t0))≤0, then the last term on the RHS is given by, 1 n/parenleftbigg −/integraldisplay M(R+|∇f|2/parenrightbigg e−fdV)/parenleftbigg1 V/integraldisplay MRdV/parenrightbigg ≥1 n/parenleftbigg −/integraldisplay M(R+|∇f|2)e−fdV/parenrightbigg/parenleftbigg/integraldisplay M(R+|∇f|2)e−fdV/parenrightbigg =−1 n/parenleftbigg/integraldisplay M(R+ ∆f)e−fdV/parenrightbigg2 . Thus att=t0, d dt¯λ(gij(t)) (1.5.8) ≥2V2 n/bracketleftbigg/integraldisplay M|Rij+∇i∇jf−1 n(R+ ∆f)gij|2e−fdV +1 n/parenleftigg/integraldisplay M(R+ ∆f)2e−fdV−/parenleftbigg/integraldisplay M(R+ ∆f)e−fdV/parenrightbigg2/parenrightigg/bracketrightbigg ≥0 by the Cauchy-Schwarz inequality. Thus we have proved state ment (i). We note that on an expanding breather on [ t1,t2] withαgij(t1) andgij(t2) diﬀer only by a diﬀeomorphism for some α>1, it would necessary have dV dt>0,for somet∈[t1,t2]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 205 On the other hand, for every t, −d dtlogV=1 V/integraldisplay MRdV≥λ(gij(t)) by the deﬁnition of λ(gij(t)). It follows that on an expanding breather on [ t1,t2], ¯λ(gij(t)) =λ(gij(t))V2 n(gij(t))<0 for somet∈[t1,t2]. Then by using statement (i), it implies ¯λ(gij(t1))<¯λ(gij(t2)) unless we are on an expanding gradient soliton. We also note t hat¯λ(gij(t)) is invariant under diﬀeomorphism and scaling which implies ¯λ(gij(t1)) =¯λ(gij(t2)). Therefore the breather must be an expanding gradient solito n. In particular part (ii) of Corollaries 1.5.4 and 1.5.5 imply that all compact steady or expanding Ricci solitons are gradient ones. Combining th is fact with Proposition (1.1.1), we immediately get Proposition 1.5.6. On a compact manifold, a steady or expanding breather is necessarily an Einstein metric. In order to handle the shrinking case, we introduce the follo wing important func- tional, also due to Perelman [103], (1.5.9) W(gij,f,τ) =/integraldisplay M[τ(R+|∇f|2) +f−n](4πτ)−n 2e−fdV wheregijis a Riemannian metric, fis a smooth function on M, andτis a positive scale parameter. Clearly the functional Wis invariant under simultaneous scaling of τandgij(or equivalently the parabolic scaling), and invariant und er diﬀeomorphism. Namely, for any positive number aand any diﬀeomorphism ϕ (1.5.10) W(aϕ∗gij,ϕ∗f,aτ) =W(gij,f,τ). Similar to Lemma 1.5.2, we have the following ﬁrst variation formula for W. Lemma 1.5.7 ( Perelman [103] ).Ifvij=δgij, h=δf,andη=δτ, then δW(vij,h,η) =/integraldisplay M−τvij/parenleftbigg Rij+∇i∇jf−1 2τgij/parenrightbigg (4πτ)−n 2e−fdV +/integraldisplay M/parenleftigv 2−h−n 2τη/parenrightig [τ(R+ 2∆f− |∇f|2) +f−n−1](4πτ)−n 2e−fdV +/integraldisplay Mη/parenleftig R+|∇f|2−n 2τ/parenrightig (4πτ)−n 2e−fdV. Herev=gijvijas before. 206 H.-D. CAO AND X.-P. ZHU Proof. Arguing as in the proof of Lemma 1.5.2, the ﬁrst variation of t he functional Wcan be computed as follows, δW(vij,h,η) =/integraldisplay M[η(R+|∇f|2) +τ(−∆v+∇i∇jvij−Rijvij−vij∇if∇jf + 2/a\}b∇acketle{t∇f,∇h/a\}b∇acket∇i}ht) +h](4πτ)−n 2e−fdV +/integraldisplay M/bracketleftig (τ(R+|∇f|2) +f−n)/parenleftig −n 2η τ+v 2−h/parenrightig/bracketrightig (4πτ)−n 2e−fdV =/integraldisplay M[η(R+|∇f|2) +h](4πτ)−n 2e−fdV +/integraldisplay M[−τvij(Rij+∇i∇jf) +τ(v−2h)(∆f− |∇f|2)](4πτ)−n 2e−fdV +/integraldisplay M/bracketleftig (τ(R+|∇f|2) +f−n)/parenleftig −n 2η τ+v 2−h/parenrightig/bracketrightig (4πτ)−n 2e−fdV =−/integraldisplay Mτvij/parenleftbigg Rij+∇i∇jf−1 2τgij/parenrightbigg (4πτ)−n 2e−fdV +/integraldisplay M/parenleftigv 2−h−n 2τη/parenrightig [τ(R+|∇f|2) +f−n+ 2τ(∆f− |∇f|2)](4πτ)−n 2e−fdV +/integraldisplay M/bracketleftig η/parenleftig R+|∇f|2−n 2τ/parenrightig +/parenleftig h−v 2+n 2τη/parenrightig/bracketrightig (4πτ)−n 2e−fdV =/integraldisplay M−τvij/parenleftbigg Rij+∇i∇jf−1 2τgij/parenrightbigg (4πτ)−n 2e−fdV +/integraldisplay M/parenleftigv 2−h−n 2τη/parenrightig [τ(R+ 2∆f− |∇f|2) +f−n−1](4πτ)−n 2e−fdV +/integraldisplay Mη/parenleftig R+|∇f|2−n 2τ/parenrightig (4πτ)−n 2e−fdV. The following result is analogous to Proposition 1.5.3. Proposition 1.5.8. Ifgij(t),f(t)andτ(t)evolve according to the system   ∂gij ∂t=−2Rij, ∂f ∂t=−∆f+|∇f|2−R+n 2τ, ∂τ ∂t=−1, then we have the identity d dtW(gij(t),f(t),τ(t)) =/integraldisplay M2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (4πτ)−n 2e−fdV and/integraltext M(4πτ)−n 2e−fdVis constant. In particular W(gij(t),f(t),τ(t))is nondecreasing in time and the monotonicity is strict unless we are on a shrin king gradient soliton. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 207 Proof. Using Lemma 1.5.7, we have d dtW(gij(t),f(t),τ(t)) (1.5.11) =/integraldisplay M2τRij/parenleftbigg Rij+∇i∇jf−1 2τgij/parenrightbigg (4πτ)−n 2e−fdV +/integraldisplay M(∆f− |∇f|2)[τ(R+ 2∆f− |∇f|2) +f](4πτ)−n 2e−fdV −/integraldisplay M/parenleftig R+|∇f|2−n 2τ/parenrightig (4πτ)−n 2e−fdV. Here we have used the fact that/integraltext M(∆f− |∇f|2)e−fdV= 0. The second term on the RHS of (1.5.11) is /integraldisplay M(∆f− |∇f|2)[τ(R+ 2∆f− |∇f|2) +f](4πτ)−n 2e−fdV =/integraldisplay M(∆f− |∇f|2)(2τ∆f−τ|∇f|2)(4πτ)−n 2e−fdV −/integraldisplay M|∇f|2(4πτ)−n 2e−fdV+τ/integraldisplay M(−∇if)(∇iR)(4πτ)−n 2e−fdV =τ/integraldisplay M(−∇if)(∇i(2∆f− |∇f|2))(4πτ)−n 2e−fdV −/integraldisplay M∆f(4πτ)−n 2e−fdV−2τ/integraldisplay M∇if∇jRij(4πτ)−n 2e−fdV =−2τ/integraldisplay M(∇if)(∇i∆f− /a\}b∇acketle{t∇f,∇i∇f/a\}b∇acket∇i}ht)(4πτ)−n 2e−fdV + 2τ/integraldisplay M[(∇i∇jf)Rij− ∇ if∇jfRij](4πτ)−n 2e−fdV + 2τ/integraldisplay M/parenleftbigg −1 2τgij/parenrightbigg (∇i∇jf)(4πτ)−n 2e−fdV =−2τ/integraldisplay M[(∇if∇jf− ∇ i∇jf)∇i∇jf−Rij∇if∇jf − ∇ i∇jf∇if∇jf](4πτ)−n 2e−fdV + 2τ/integraldisplay M[(∇i∇jf)Rij− ∇ if∇jfRij](4πτ)−n 2e−fdV + 2τ/integraldisplay M/parenleftbigg −1 2τgij/parenrightbigg (∇i∇jf)(4πτ)−n 2e−fdV = 2τ/integraldisplay M(∇i∇jf)/parenleftbigg ∇i∇jf+Rij−1 2τgij/parenrightbigg (4πτ)−n 2e−fdV. Also the third term on the RHS of (1.5.11) is /integraldisplay M−/parenleftig R+|∇f|2−n 2τ/parenrightig (4πτ)−n 2e−fdV =/integraldisplay M−/parenleftig R+ ∆f−n 2τ/parenrightig (4πτ)−n 2e−fdV = 2τ/integraldisplay M/parenleftbigg−1 2τgij/parenrightbigg/parenleftbigg Rij+∇i∇jf−1 2τgij/parenrightbigg (4πτ)−n 2e−fdV. 208 H.-D. CAO AND X.-P. ZHU Therefore, by combining the above identities, we obtain d dtW(gij(t),f(t),τ(t)) = 2τ/integraldisplay M/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (4πτ)−n 2e−fdV. Finally, by using the computations in (1.5.5) and the evolut ion equations of f andτ, we have ∂ ∂t/parenleftbig (4πτ)−n 2e−fdV/parenrightbig = (4πτ)−n 2/bracketleftbigg∂ ∂t(e−fdV) +n 2τe−fdV/bracketrightbigg =−(4πτ)−n 2∆(e−f)dV. Hence d dt/integraldisplay M(4πτ)−n 2e−fdV=−(4πτ)−n 2/integraldisplay M∆(e−f)dV= 0. Now we set (1.5.12) µ(gij,τ) = inf/braceleftbigg W(gij,f,τ)|f∈C∞(M),1 (4πτ)n/2/integraldisplay Me−fdV= 1/bracerightbigg and ν(gij) = inf/braceleftbigg W(g,f,τ)|f∈C∞(M),τ >0,1 (4πτ)n/2/integraldisplay e−fdV= 1/bracerightbigg . Note that if we let u=e−f/2, then the functional Wcan be expressed as W(gij,f,τ) =/integraldisplay M[τ(Ru2+ 4|∇u|2)−u2logu2−nu2](4πτ)−n 2dV and the constraint/integraltext M(4πτ)−n 2e−fdV= 1 becomes/integraltext Mu2(4πτ)−n 2dV= 1.Thus µ(gij,τ) corresponds to the best constant of a logarithmic Sobolev i nequality. Since the nonquadratic term is subcritical (in view of Sobolev exp onent), it is rather straightforward to show that inf/braceleftbigg/integraldisplay M[τ(4|∇u|2+Ru2)−u2logu2−nu2](4πτ)−n 2dV/vextendsingle/vextendsingle/vextendsingle/integraldisplay Mu2(4πτ)−n 2dV=1/bracerightbigg is achieved by some nonnegative function u∈H1(M) which satisﬁes the Euler- Lagrange equation τ(−4∆u+Ru)−2ulogu−nu=µ(gij,τ)u. One can further show that uis positive (see [108]). Then the standard regularity theory of elliptic PDEs shows that uis smooth. We refer the reader to Rothaus [108] for more details. It follows that µ(gij,τ) is achieved by a minimizer fsatisfying the nonlinear equation (1.5.13) τ(2∆f− |∇f|2+R) +f−n=µ(gij,τ). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 209 Corollary 1.5.9. (i)µ(gij(t),τ−t)is nondecreasing along the Ricci ﬂow; moveover, the monoton ic- ity is strict unless we are on a shrinking gradient soliton; (ii) A shrinking breather is necessarily a shrinking gradie nt soliton. Proof. Fix any time t0, letf0be a minimizer of µ(gij(t0),τ−t0).Note that the backward heat equation ∂f ∂t=−∆f+|∇f|2−R+n 2τ is equivalent to the linear equation ∂ ∂t((4πτ)−n 2e−f) =−∆((4πτ)−n 2e−f) +R((4πτ)−n 2e−f). Thus we can solve the backward heat equation of fwithf|t=t0=f0to obtain f(t),t≤t0, with/integraltext M(4πτ)−n 2e−f(t)dV= 1.It then follows from Proposition 1.5.8 that µ(gij(t),τ−t)≤ W(gij(t),f(t),τ−t) ≤ W(gij(t0),f(t0),τ−t0) =µ(gij(t0),τ−t0) fort≤t0and the second inequality is strict unless we are on a shrinki ng gradient soliton. This proves statement (i). Consider a shrinking breather on [ t1,t2] withαgij(t1) andgij(t2) diﬀer only by a diﬀeomorphism for some α <1.Recall that the functional Wis invariant under simultaneous scaling of τandgijand invariant under diﬀeomorphism. Then for τ >0 to be determined, µ(gij(t1),τ−t1) =µ(αgij(t1),α(τ−t1)) =µ(gij(t2),α(τ−t1)) and by the monotonicity of µ(gij(t),τ−t), µ(gij(t1),τ−t1)≤µ(gij(t2),τ−t2). Now takeτ >0 such that α(τ−t1) =τ−t2, i.e., τ=t2−αt1 1−α. This shows the equality holds in the monotonicity of µ(gij(t),τ−t).So the shrinking breather must be a shrinking gradient soliton. Finally, we remark that Hamilton, Ilmanen and the ﬁrst autho r [18] have obtained the second variation formulas for both λ-energy and ν-energy. We refer the reader to their paper [18] for more details and related stability ques tions. 210 H.-D. CAO AND X.-P. ZHU 2. Maximum Principle and Li-Yau-Hamilton Inequalities. The maxi- mum principle is a fundamental tool in the study of parabolic equations in general. In this chapter, we present various maximum principles for ten sors developed by Hamil- ton in the Ricci ﬂow. As an immediate consequence, the Ricci ﬂ ow preserves the nonnegativity of the curvature operator. We also present th e two crucial estimates in the Ricci ﬂow: the Hamilton-Ivey curvature pinching esti mate (when dimension n= 3), and the Li-Yau-Hamilton estimate from which one obtain s the Harnack in- equality for the evolved scalar curvature via a Li-Yau path i ntegral. Finally, we describe Perelman’s Li-Yau type estimate for solutions to t he conjugate heat equa- tion and show how Li-Yau type path integral leads to a space-t ime distance function (i.e., what Perelman called the reduced distance). 2.1. Preserving Positive Curvature. LetMbe ann-dimensional complete manifold. Consider a family of smooth metrics gij(t) evolving by the Ricci ﬂow with uniformly bounded curvature for t∈[0,T] withT <+∞. Denote by dt(x,y) the distance between two points x,y∈Mwith respect to the metric gij(t). Lemma 2.1.1. There exists a smooth function fonMsuch thatf≥1every- where,f(x)→+∞asd0(x,x0)→+∞(for some ﬁxed x0∈M), |∇f|gij(t)≤Cand|∇2f|gij(t)≤C onM×[0,T]for some positive constant C. Proof. Letϕ(v) be a smooth function on Rnwhich is nonnegative, rotation- ally symmetric and has compact support in a small ball center ed at the origin with/integraltext Rnϕ(v)dv= 1. For eachx∈M, set f(x) =/integraldisplay Rnϕ(v)(d0(x0,exp x(v)) + 1)dv, where the integral is taken over the tangent space TxMatxwhich we have identiﬁed with Rn. If the size of the support of ϕ(v) is small compared to the maximum curvature, then it is well known that this deﬁnes a smooth fun ctionfonMwith f(x)→+∞asd0(x,x0)→+∞, while the bounds on the ﬁrst and second covariant derivatives of fwith respect to the metric gij(·,0) follow from the Hessian comparison theorem. Thus it remains to show these bounds hold with respe ct to the evolving metricgij(t). We compute, using the frame {Fi a∇if}introduced in Section 1.3, ∂ ∂t∇af=∂ ∂t(Fi a∇if) =Rab∇bf. Hence |∇f| ≤C1·eC2t, whereC1,C2are some positive constants depending only on the dimension . Also ∂ ∂t(∇a∇bf) =∂ ∂t/parenleftbigg Fi aFj b/parenleftbigg∂2f ∂xi∂xj−Γk ij∂f ∂xk/parenrightbigg/parenrightbigg =Rac∇b∇cf+Rbc∇a∇cf+ (∇cRab− ∇ aRbc− ∇ bRac)∇cf. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 211 Then by Shi’s derivative estimate (Theorem 1.4.1), we have ∂ ∂t|∇2f| ≤C3|∇2f|+C3√ t, which implies |∇2f|gij(t)≤eC3t/parenleftbigg |∇2f|gij(0)+/integraldisplayt 0C3√τe−C3τdτ/parenrightbigg for some positive constants C3depending only on the dimension and the curvature bound. We now use the weak maximum principle to derive the following Proposition 2.1.2. If the scalar curvature Rof the solution gij(t),0≤t≤T, to the Ricci ﬂow is nonnegative at t= 0, then it remains so on 0≤t≤T. Proof. Letfbe the function constructed in Lemma 2.1.1 and recall ∂R ∂t= ∆R+ 2|Ric|2. For any small constant ε>0 and large constant A>0, we have ∂ ∂t(R+εeAtf) =∂R ∂t+εAeAtf = ∆(R+εeAtf) + 2|Ric|2+εeAt(Af−∆f) >∆(R+εeAtf) by choosing Alarge enough. We claim that R+εeAtf >0 onM×[0,T]. Suppose not, then there exist a ﬁrst time t0>0 and a point x0∈Msuch that (R+εeAtf)(x0,t0) = 0, ∇(R+εeAtf)(x0,t0) = 0, ∆(R+εeAtf)(x0,t0)≥0, and∂ ∂t(R+εeAtf)(x0,t0)≤0. Then 0≥∂ ∂t(R+εeAtf)(x0,t0)>∆(R+εeAtf)(x0,t0)≥0, which is a contradiction. So we have proved that R+εeAtf >0 onM×[0,T]. Lettingε→0, we get R≥0 onM×[0,T]. 212 H.-D. CAO AND X.-P. ZHU This ﬁnishes the proof of the proposition. Next we derive a maximum principle of Hamilton for tensors. L etMbe a complete manifold with a metric g={gij},Va vector bundle over Mwith a metric h={hαβ} and a connection ∇={Γα iβ}compatible with h, and suppose his ﬁxed but gand∇ may vary smoothly with time t. Let Γ(V) be the vector space of C∞sections of V. The Laplacian ∆ acting on a section σ∈Γ(V) is deﬁned by ∆σ=gij∇i∇jσ. LetMαβbe a symmetric bilinear form on V. We sayMαβ≥0 ifMαβvαvβ≥0 for all vectorsv={vα}. AssumeNαβ=P(Mαβ,hαβ) is a polynomial in Mαβformed by contracting products of Mαβwith itself using the metric h={hαβ}. Assume that the tensorMαβis uniformly bounded in space-time and let gijevolve by the Ricci ﬂow with bounded curvature. Lemma 2.1.3. Suppose that on 0≤t≤T, ∂ ∂tMαβ= ∆Mαβ+ui∇iMαβ+Nαβ whereui(t)is a time-dependent vector ﬁeld on Mwith uniform bound and Nαβ= P(Mαβ,hαβ)satisﬁes Nαβvαvβ≥0whenever Mαβvβ= 0. IfMαβ≥0att= 0, then it remains so on 0≤t≤T. Proof. Set ˜Mαβ=Mαβ+εeAtfhαβ, whereA >0 is a suitably large constant (to be chosen later) and fis the function constructed in Lemma 2.1.1. We claim that ˜Mαβ>0 onM×[0,T] for everyε>0. If not, then for some ε>0, there will be a ﬁrst time t0>0 where ˜Mαβacquires a null vector vαof unit length at some point x0∈M. At (x0,t0), Nαβvαvβ≥Nαβvαvβ−˜Nαβvαvβ ≥ −CεeAt0f(x0), where ˜Nαβ=P(˜Mαβ,hαβ), andCis a positive constant (depending on the bound of Mαβ, but independent of A). Let us extend vαto a local vector ﬁeld in a neighborhood of x0by parallel trans- latingvαalong geodesics (with respect to the metric gij(t0)) emanating radially out ofx0, withvαindependent of t. Then, at ( x0,t0), we have ∂ ∂t(˜Mαβvαvβ)≤0, ∇(˜Mαβvαvβ) = 0, and ∆( ˜Mαβvαvβ)≥0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 213 But 0≥∂ ∂t(˜Mαβvαvβ) =∂ ∂t(Mαβvαvβ+εeAtf), = ∆( ˜Mαβvαvβ)−∆(εeAtf) +ui∇i(˜Mαβvαvβ) −ui∇i(εeAtf) +Nαβvαvβ+εAeAt0f(x0) ≥ −CεeAt0f(x0) +εAeAt0f(x0)>0 whenAis chosen suﬃciently large. This is a contradiction. By applying Lemma 2.1.3 to the evolution equation ∂ ∂tMαβ= ∆Mαβ+M2 αβ+M# αβ of the curvature operator Mαβ, we immediately obtain the following important result. Proposition 2.1.4 ( Hamilton [59] ).Nonnegativity of the curvature operator Mαβis preserved by the Ricci ﬂow. In the K¨ ahler case, the nonnegativity of the holomorpic bis ectional curvature is preserved under the K¨ ahler-Ricci ﬂow. This result is prove d by Bando [5] for complex dimensionn= 3 and by Mok [92] for general dimension nwhen the manifold is compact, and by Shi [115] when the manifold is noncompact. Proposition 2.1.5. Under the K¨ ahler-Ricci ﬂow if the initial metric has posi- tive(nonnegative )holomorphic bisectional curvature then the evolved metric also has positive (nonnegative )holomorphic bisectional curvature. 2.2. Strong Maximum Principle. Let Ω be a bounded, connected open set of a complete n-dimensional manifold M, and letgij(x,t) be a smooth solution to the Ricci ﬂow on Ω ×[0,T]. Consider a vector bundle Vover Ω with a ﬁxed metric hαβ (independent of time), and a connection ∇={Γα iβ}which is compatible with hαβand may vary with time t. Let Γ(V) be the vector space of C∞sections ofVover Ω. The Laplacian ∆ acting on a section σ∈Γ(V) is deﬁned by ∆σ=gij(x,t)∇i∇jσ. Consider a family of smooth symmetric bilinear forms Mαβevolving by (2.2.1)∂ ∂tMαβ= ∆Mαβ+Nαβ,on Ω×[0,T], whereNαβ=P(Mαβ,hαβ) is a polynomial in Mαβformed by contracting products ofMαβwith itself using the metric hαβand satisﬁes Nαβ≥0,wheneverMαβ≥0. The following result, due to Hamilton [59], shows that the so lution of (2.2.1) satisﬁes a strong maximum principle. Theorem 2.2.1 ( Hamilton’s strong maximum principle ).LetMαβbe a smooth solution of the equation (2.2.1). SupposeMαβ≥0onΩ×[0,T]. Then there exists a positive constant 0<δ≤Tsuch that on Ω×(0,δ), the rank of Mαβis constant, and 214 H.-D. CAO AND X.-P. ZHU the null space of Mαβis invariant under parallel translation and invariant in ti me and also lies in the null space of Nαβ. Proof. Set l= max x∈Ω{rank ofMαβ(x,0)}. Then we can ﬁnd a nonnegative smooth function ρ(x), which is positive somewhere and has compact support in Ω, so that at every point x∈Ω, n−l+1/summationdisplay i=1Mαβ(x,0)vα ivβ i≥ρ(x) for any (n−l+ 1) orthogonal unit vectors {v1,...,v n−l+1}atx. Let us evolve ρ(x) by the heat equation ∂ ∂tρ= ∆ρ with the Dirichlet condition ρ|∂Ω= 0 to get a smooth function ρ(x,t) deﬁned on Ω×[0,T]. By the standard strong maximum principle, we know that ρ(x,t) is positive everywhere in Ω for all t∈(0,T]. For everyε>0, we claim that at every point ( x,t)∈Ω×[0,T], there holds n−l+1/summationdisplay i=1Mαβ(x,t)vα ivβ i+εet>ρ(x,t) for any (n−l+ 1) orthogonal unit vectors {v1,...,v n−l+1}atx. We argue by contradiction. Suppose not, then for some ε>0, there will be a ﬁrst timet0>0 and some ( n−l+ 1) orthogonal unit vectors {v1,...,v n−l+1}at some pointx0∈Ω so that n−l+1/summationdisplay i=1Mαβ(x0,t0)vα ivβ i+εet0=ρ(x0,t0) Let us extend each vi(i= 1,...,n −l+ 1) to a local vector ﬁeld, independent oft, in a neighborhood of x0by parallel translation along geodesics (with respect to the metric gij(t0)) emanating radially out of x0. Clearly {v1,...,v n−l+1}remain orthogonal unit vectors in the neighborhood. Then, at ( x0,t0), we have ∂ ∂t/parenleftiggn−l+1/summationdisplay i=1Mαβvα ivβ i+εet−ρ/parenrightigg ≤0, and ∆/parenleftiggn−l+1/summationdisplay i=1Mαβvα ivβ i+εet−ρ/parenrightigg ≥0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 215 But, sinceNαβ≥0 by our assumption, we have 0≥∂ ∂t/parenleftiggn−l+1/summationdisplay i=1Mαβvα ivβ i+εet−ρ/parenrightigg =n−l+1/summationdisplay i=1(∆Mαβ+Nαβ)vα ivβ i+εet−∆ρ ≥n−l+1/summationdisplay i=1∆(Mαβvα ivβ i) +εet−∆ρ =n−l+1/summationdisplay i=1∆(Mαβvα ivβ i+εet−ρ) +εet ≥εet>0. This is a contradiction. Thus by letting ε→0, we prove that n−l+1/summationdisplay i=1Mαβ(x,t)vα ivβ i≥ρ(x,t) for any (n−l+ 1) orthogonal unit vectors {v1,...,v n−l+1}atx∈Ω andt∈[0,T]. HenceMαβhas at least rank leverywhere in the open set Ω for all t∈(0,T]. Therefore we can ﬁnd a positive constant δ(≤T) such that the rank Mαβis constant over Ω×(0,δ). Next we proceed to analyze the null space of Mαβ. Letvbe any smooth section ofVin the null of Mαβon 0<t<δ . Then 0 =∂ ∂t(Mαβvαvβ) =/parenleftbigg∂ ∂tMαβ/parenrightbigg vαvβ+ 2Mαβvα∂vβ ∂t =/parenleftbigg∂ ∂tMαβ/parenrightbigg vαvβ, and 0 = ∆(Mαβvαvβ) = (∆Mαβ)vαvβ+ 4gkl∇kMαβ·vα∇lvβ + 2Mαβgkl∇kvα· ∇lvβ+ 2Mαβvα∆vβ = (∆Mαβ)vαvβ+ 4gkl∇kMαβ·vα∇lvβ+ 2Mαβgkl∇kvα· ∇lvβ. By noting that 0 =∇k(Mαβvβ) = (∇kMαβ)vα+Mαβ∇kvα and using the evolution equation (2.2.1), we get Nαβvαvβ+ 2Mαβgkl∇kvα· ∇lvβ= 0. 216 H.-D. CAO AND X.-P. ZHU SinceMαβ≥0 andNαβ≥0, we must have v∈null (Nαβ) and ∇iv∈null (Mαβ),for alli. The ﬁrst inclusion shows that null ( Mαβ)⊂null (Nαβ),and the second inclusion shows that null ( Mαβ) is invariant under parallel translation. To see null ( Mαβ) is also invariant in time, we ﬁrst note that ∆v=∇i(∇iv)∈null (Mαβ) and then gkl∇kMαβ· ∇lvα=gkl∇k(Mαβ∇lvα)−Mαβ∆vα= 0. Thus we have 0 = ∆(Mαβvα) = (∆Mαβ)vα+ 2gkl∇kMαβ· ∇lvα+Mαβ∆vα = (∆Mαβ)vα, and hence 0 =∂ ∂t(Mαβvα) = (∆Mαβ+Nαβ)vα+Mαβ∂vα ∂t =Mαβ∂vα ∂t. This shows that ∂v ∂t∈null (Mαβ), so the null space of Mαβis invariant in time. We now apply Hamilton’s strong maximum principle to the evol ution equation of the curvature operator Mαβ. Recall ∂Mαβ ∂t= ∆Mαβ+M2 αβ+M# αβ whereM# αβ=Cξγ αCηθ βMξηMγθ. Suppose we have a solution to the Ricci ﬂow with nonnegative curvature operator. Then by Theorem 2.2.1, the null space of the cur- vature operator Mαβof the solution has constant rank and is invariant in time and under parallel translation over some time interval 0 <t<δ . Moreover the null space ofMαβmust also lie in the null space of M# αβ. Denote by ( n−k) the rank of Mαβon 0<t<δ . Let us diagonalize Mαβso that Mαα= 0 ifα≤kandMαα>0 ifα>k . Then we have M# αα= 0 also for α≤kfrom the evolution equation of Mαα. Since 0 =M# αα=Cξγ αCηθ αMξηMγθ, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 217 it follows that Cξγ α=/a\}b∇acketle{tvα,[vξ,vγ]/a\}b∇acket∇i}ht = 0,ifα≤kandξ,γ>k. This says that the image of Mαβis a Lie subalgebra (in fact it is the subalgebra of the restricted holonomy group by using the Ambrose-Singer h olonomy theorem [3]). This proves the following result. Theorem 2.2.2 ( Hamilton [59] ).Suppose the curvature operator Mαβof the initial metric is nonnegative. Then, under the Ricci ﬂow, fo r some interval 0<t<δ the image of Mαβis a Lie subalgebra of so(n)which has constant rank and is invariant under parallel translation and invariant in time. 2.3. Advanced Maximum Principle for Tensors. In this section we present Hamilton’s advanced maximum principle for tensors which ge neralizes Lemma 2.1.3 and shows how a tensor evolving by a nonlinear heat equation m ay be controlled by a system of ODEs. An important application of the advanced ma ximum principle is the Hamilton-Ivey curvature pinching estimate for the Ricc i ﬂow on three-manifolds given in the next section. More applications will be given in Chapter 5. LetMbe a complete manifold equipped with a one-parameter family of Rie- mannian metrics gij(t), 0≤t≤T, withT <+∞. LetV→Mbe a vector bundle with a time-independent bundle metric haband Γ(V) be the vector space of C∞ sections ofV. Let ∇t: Γ(V)→Γ(V⊗T∗M), t∈[0,T] be a smooth family of time-dependent connections compatibl e withhab, i.e. (∇t)ihab∆= (∇t)∂ ∂xihab= 0, for any local coordinate {∂ ∂x1,...,∂ ∂xn}.The Laplacian ∆ tacting on a section σ∈ Γ(V) is deﬁned by ∆tσ=gij(x,t)(∇t)i(∇t)jσ. For the application to the Ricci ﬂow, we will always assume th at the metrics gij(·,t) evolve by the Ricci ﬂow. Since Mmay be noncompact, we assume that, for the sake of simplicity, the curvature of gij(t) is uniformly bounded on M×[0,T]. LetN:V×[0,T]→Vbe a ﬁber preserving map, i.e., N(x,σ,t) is a time- dependent vector ﬁeld deﬁned on the bundle Vand tangent to the ﬁbers. We assume thatN(x,σ,t) is continuous in x,tand satisﬁes |N(x,σ1,t)−N(x,σ2,t)| ≤CB|σ1−σ2| for allx∈M,t∈[0,T] and |σ1| ≤B,|σ2| ≤B, whereCBis a positive constant depending only on B. Then we can form the nonlinear heat equation (PDE)∂ ∂tσ(x,t) = ∆ tσ(x,t) +ui(∇t)iσ(x,t) +N(x,σ(x,t),t) whereui=ui(t) is a time-dependent vector ﬁeld on Mwhich is uniformly bounded onM×[0,T]. LetKbe a closed subset of V. One important question is under what conditions will solutions of the PDE which start in Kremain inK. To answer this question, Hamilton [59] imposed the following two conditio ns onK: 218 H.-D. CAO AND X.-P. ZHU (H1)Kis invariant under parallel translation deﬁned by the conne ction∇tfor eacht∈[0,T]; (H2) in each ﬁber Vx, the setKx∆=Vx∩Kis closed and convex. Then one can judge the behavior of the PDE by comparing to that of the following ODE (ODE)dσx dt=N(x,σx,t) forσx=σx(t) in each ﬁber Vx. Theorem 2.3.1 ( Hamilton’s advanced maximum principle [59] ).LetKbe a closed subset of Vsatisfying the hypothesis (H1)and(H2). Suppose that for any x∈Mand any initial time t0∈[0,T), any solution σx(t)of the (ODE) which starts inKxatt0will remain in Kxfor all later times. Then for any initial time t0∈[0,T) the solution σ(x,t)of the (PDE) will remain in Kfor all later times provided σ(x,t) starts inKat timet0andσ(x,t)is uniformly bounded with respect to the bundle metrichabonM×[t0,T]. We remark that Lemma 2.1.3 is a special case of the above theor em whereVis given by a symmetric tensor product of a vector bundle and Kcorresponds to the convex set consisting of all nonnegative symmetric bilinea r forms. We also remark that Hamilton [59] established the above theorem for a gener al evolving metric gij(x,t) which does not necessarily satisfy the Ricci ﬂow. Before proving Theorem 2.3.1, we need to establish three lem mas. Letϕ: [a,b]→ Rbe a Lipschitz function. We considerdϕ dt(t) att∈[a,b) in the sense of limsup of the forward diﬀerence quotients, i.e., dϕ dt(t) = limsup h→0+ϕ(t+h)−ϕ(t) h. Lemma 2.3.2. Supposeϕ: [a,b]→Ris Lipschitz continuous and suppose for some constant C <+∞, d dtϕ(t)≤Cϕ(t), whenever ϕ (t)≥0on[a,b), andϕ(a)≤0. Thenϕ(t)≤0on[a,b]. Proof. By replacing ϕbye−Ctϕ, we may assume d dtϕ(t)≤0,whenever ϕ(t)≥0 on [a,b), andϕ(a)≤0. For arbitrary ε >0, we shall show ϕ(t)≤ε(t−a) on [a,b]. Clearly we may assume ϕ(a) = 0. Since limsup h→0+ϕ(a+h)−ϕ(a) h≤0, there must be some interval a≤t<δ on whichϕ(t)≤ε(t−a). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 219 Leta≤t<cbe the largest interval with c≤bsuch thatϕ(t)≤ε(t−a) on [a,c). Then by continuity ϕ(t)≤ε(t−a) on the closed interval [ a,c]. We claim that c=b. Suppose not, then we can ﬁnd δ>0 such that ϕ(t)≤ε(t−a) on [a,c+δ] since limsup h→0+ϕ(c+h)−ϕ(c) h≤0. This contradicts the choice of the largest interval [ a,c). Therefore, since ε>0 can be arbitrary small, we have proved ϕ(t)≤0 on [a,b]. The second lemma below is a general principle on the derivati ve of a sup-function which will bridge solutions between ODEs and PDEs. Let Xbe a complete smooth manifold and Ybe a compact subset of X. Letψ(x,t) be a smooth function on X×[a,b] and letϕ(t) = sup {ψ(y,t)|y∈Y}. Then it is clear that ϕ(t) is Lipschitz continuous. We have the following useful estimate on its der ivative. Lemma 2.3.3. d dtϕ(t)≤sup/braceleftbigg∂ψ ∂t(y,t)|y∈Ysatisﬁesψ(y,t) =ϕ(t)/bracerightbigg . Proof. Choose a sequence of times {tj}decreasing to tfor which lim tj→tϕ(tj)−ϕ(t) tj−t=dϕ(t) dt. SinceYis compact, we can choose yj∈Ywithϕ(tj) =ψ(yj,tj). By passing to a subsequence, we can assume yj→yfor somey∈Y. By continuity, we have ϕ(t) =ψ(y,t). It follows that ψ(yj,t)≤ψ(y,t), and then ϕ(tj)−ϕ(t)≤ψ(yj,tj)−ψ(yj,t) =∂ ∂tψ(yj,˜tj)·(tj−t) for some ˜tj∈[t,tj] by the mean value theorem. Thus we have lim tj→tϕ(tj)−ϕ(t) tj−t≤∂ ∂tψ(y,t). This proves the result. We remark that the above two lemmas are somewhat standard fac ts in the theory of PDEs and we have implicitly used them in the previous secti ons when we apply the maximum principle. The third lemma gives a characteriza tion of when a system of ODEs preserve closed convex sets in Euclidean space. Let Z⊂Rnbe a closed convex subset. We deﬁne the tangent cone TϕZto the closed convex set Zat a pointϕ∈∂Zas the smallest closed convex cone with vertex at ϕwhich contains Z. Lemma 2.3.4. LetU⊂Rnbe an open set and Z⊂Ube a closed convex subset. Consider the ODE (2.3.1)dϕ dt=N(ϕ,t) whereN:U×[0,T]→Rnis continuous and Lipschitz in ϕ. Then the following two statements are equivalent. 220 H.-D. CAO AND X.-P. ZHU (i) For any initial time t0∈[0,T], any solution of the ODE (2.3.1)which starts in Z att0will remain in Z for all later times; (ii)ϕ+N(ϕ,t)∈TϕZfor allϕ∈∂Zandt∈[0,T). Proof. We say that a linear function lonRnis asupport function forZ atϕ∈∂Zand writel∈SϕZif|l|= 1 andl(ϕ)≥l(η) for allη∈Z. Then ϕ+N(ϕ,t)∈TϕZif and only if l(N(ϕ,t))≤0 for alll∈SϕZ. Supposel(N(ϕ,t))>0 for someϕ∈∂Zand somel∈SϕZ.Then d dtl(ϕ) =l/parenleftbiggdϕ dt/parenrightbigg =l(N(ϕ,t))>0, sol(ϕ) is strictly increasing and the solution ϕ(t) of the ODE (2.3.1) cannot remain inZ. To see the converse, ﬁrst note that we may assume Zis compact. This is because we can modify the vector ﬁeld N(ϕ,t) by multiplying a cutoﬀ function which is every- where nonnegative, equals one on a large ball and equals zero on the complement of a larger ball. The paths of solutions of the ODE are unchanged inside the ﬁrst large ball, so we can intersect Zwith the second ball to make Zconvex and compact. If there were a counterexample before the modiﬁcation there wo uld still be one after as we chose the ﬁrst ball large enough. Lets(ϕ) be the distance from ϕtoZinRn. Clearlys(ϕ) = 0 ifϕ∈Z. Then s(ϕ) = sup {l(ϕ−η)|η∈∂Zandl∈SηZ}. The sup is taken over a compact subset of Rn×Rn. Hence by Lemma 2.3.3 d dts(ϕ)≤sup{l(N(ϕ,t))|η∈∂Z,l∈SηZands(ϕ) =l(ϕ−η)}. It is clear that the sup on the RHS of the above inequality can b e takeen only when ηis the unique closest point in Ztoϕandlis the linear function of length one with gradient in the direction of ϕ−η. SinceN(ϕ,t) is Lipschitz in ϕand continuous in t, we have |N(ϕ,t)−N(η,t)| ≤C|ϕ−η| for some constant Cand allϕandηin the compact set Z. By hypothesis (ii), l(N(η,t))≤0, and for the unique η, the closest point in Ztoϕ, |ϕ−η|=s(ϕ). Thus d dts(ϕ)≤sup/braceleftiggl(N(η,t)) +|l(N(ϕ,t))−l(N(η,t))| |η∈∂Z, l∈SηZ,ands(ϕ) =l(ϕ−η)/bracerightigg ≤Cs(ϕ). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 221 Sinces(ϕ) = 0 to start at t0, it follows from Lemma 2.3.2 that s(ϕ) = 0 fort∈[t0,T]. This proves the lemma. We are now ready to prove Theorem 2.3.1. Proof of Theorem 2.3.1. Since the solution σ(x,t) of the (PDE) is uniformly bounded with respect to the bundle metric habonM×[t0,T] by hypothesis, we may assume that Kis contained in a tubular neighborhood V(r) of the zero section in V whose intersection with each ﬁber Vxis a ball of radius raround the origin measured by the bundle metric habfor some large r>0. Recall that gij(·,t),t∈[0,T], is a smooth solution to the Ricci ﬂow with uniformly bounded curvature on M×[0,T]. From Lemma 2.1.1, we have a smooth function f such thatf≥1 everywhere, f(x)→+∞asd0(x,x0)→+∞for some ﬁxed point x0∈M, and the ﬁrst and second covariant derivatives with respect to the metrics gij(·,t) are uniformly bounded on M×[0,T]. Using the metric habin each ﬁber Vx and writing |ϕ−η|for the distance between ϕ∈Vxandη∈Vx, we set s(t) = sup x∈M{inf{|σ(x,t)−η| |η∈Kx∆=K∩Vx} −ǫeAtf(x)} whereǫis an arbitrarily small positive number and Ais a positive constant to be determined. We rewrite the function s(t) as s(t) = sup {l(σ(x,t)−η)−ǫeAtf(x)|x∈M,η∈∂Kxandl∈SηKx}. By the construction of the function f, we see that the sup is taken in a compact subset ofM×V×V∗for allt. Then by Lemma 2.3.3, (2.3.2)ds(t) dt≤sup/braceleftbigg∂ ∂tl(σ(x,t)−η)−ǫAeAtf(x)/bracerightbigg where the sup is over all x∈M,η∈∂Kxandl∈SηKxsuch that l(σ(x,t)−η)−ǫeAtf(x) =s(t); in particular we have |σ(x,t)−η|=l(σ(x,t)−η), whereηis the unique closest point inKxtoσ(x,t), andlis the linear function of length one on the ﬁber Vxwith gradient in the direction of ηtoσ(x,t). We compute at these ( x,η,l), ∂ ∂tl(σ(x,t)−η)−ǫAeAtf(x) (2.3.3) =l/parenleftbigg∂σ(x,t) ∂t/parenrightbigg −ǫAeAtf(x) =l(∆tσ(x,t)) +l(ui(x,t)(∇t)iσ(x,t)) +l(N(x,σ(x,t),t))−ǫAeAtf(x). By the assumption and Lemma 2.3.4 we have η+N(x,η,t)∈TηKx. Hence, for those (x,η,l),l(N(x,η,t))≤0 and then l(N(x,σ(x,t),t)) (2.3.4) ≤l(N(x,η,t)) +|N(x,σ(x,t),t)−N(x,η,t)| ≤C|σ(x,t)−η|=C(s(t) +ǫeAtf(x)) 222 H.-D. CAO AND X.-P. ZHU for some positive constant Cby the assumption that N(x,σ,t) is Lipschitz in σand the fact that the sup is taken on a compact set. Thus the combinati on of (2.3.2)–(2.3.4) gives (2.3.5)ds(t) dt≤l(∆tσ(x,t)) +l(ui(x,t)(∇t)iσ(x,t)) +Cs(t) +ǫ(C−A)eAtf(x) for thosex∈M,η∈∂Kxandl∈SηKxsuch thatl(σ(x,t)−η)−ǫeAtf(x) =s(t). Next we estimate the ﬁrst two terms of (2.3.5). As we extend a v ector in a bundle from a point xby parallel translation along geodesics emanating radiall y out ofx, we will get a smooth section of the bundle in some small neighbor hood ofxsuch that all the symmetrized covariant derivatives at xare zero. Now let us extend η∈Vxand l∈V∗ xin this manner. Clearly, we continue to have |l|(·) = 1. Since Kis invariant under parallel translations, we continue to have η(·)∈∂Kandl(·) as a support function for Katη(·). Therefore l(σ(·,t)−η(·))−ǫeAtf(·)≤s(t) in the neighborhood. It follows that the function l(σ(·,t)−η(·))−ǫeAtf(·) has a local maximum at x, so atx (∇t)i(l(σ(x,t)−η)−ǫeAtf(x)) = 0, and ∆ t(l(σ(x,t)−η)−ǫeAtf(x))≤0. Hence atx l((∇t)iσ(x,t))−ǫeAt(∇t)if(x) = 0, andl(∆tσ(x,t))−ǫeAt∆tf(x)≤0. Therefore by combining with (2.3.5), we have d dts(t)≤Cs(t) +ǫ(∆tf(x) +ui(∇t)if(x) + (C−A)f(x))eAt ≤Cs(t) forA>0 large enough, since f(x)≥1 and the ﬁrst and second covariant derivatives offare uniformly bounded on M×[0,T]. So by applying Lemma 2.3.2 and the arbitrariness of ǫ, we have completed the proof of Theorem 2.3.1. Finally, we would like to state a useful generalization of Th eorem 2.3.1 by Chow and Lu in [40] which allows the set Kto depend on time. One can consult the paper [40] for the proof. Theorem 2.3.5 ( Chow and Lu [40] ).LetK(t)⊂V,t∈[0,T]be closed subsets which satisfy the following hypotheses (H3)K(t)is invariant under parallel translation deﬁned by the conne ction∇tfor eacht∈[0,T]; (H4) in each ﬁber Vx, the setKx(t)∆=K(t)∩Vxis nonempty, closed and convex for eacht∈[0,T]; (H5) the space-time track/uniontext t∈[0,T](∂K(t)× {t})is a closed subset of V×[0,T]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 223 Suppose that, for any x∈Mand any initial time t0∈[0,T), and for any solution σx(t)of the (ODE) which starts in Kx(t0), the solution σx(t)will remain in Kx(t) for all later times. Then for any initial time t0∈[0,T)the solution σ(x,t)of the (PDE) will remain in K(t)for all later times if σ(x,t)starts inK(t0)at timet0 and the solution σ(x,t)is uniformly bounded with respect to the bundle metric habon M×[t0,T]. 2.4. Hamilton-Ivey Curvature Pinching Estimate. The Hamilton-Ivey curvature pinching estimate roughly says that if a solution to the Ricci ﬂow on a three-manifold becomes singular (i.e., the curvature goes to inﬁnity) as time tap- proaches the maximal time T, then the most negative sectional curvature will be small compared to the most positive sectional curvature. Th is pinching estimate plays a crucial role in analyzing the formation of singulari ties in the Ricci ﬂow on three-manifolds. Consider a complete solution to the Ricci ﬂow ∂ ∂tgij=−2Rij on a complete three-manifold with bounded curvature in spac e for each time t≥0. Recall from Section 1.3 that the evolution equation of the cu rvature operator Mαβis given by (2.4.1)∂ ∂tMαβ= ∆Mαβ+M2 αβ+M# αβ whereM2 αβis the operator square M2 αβ=MαγMβγ andM# αβis the Lie algebra so(n) square M# αβ=Cγζ αCηθ βMγηMζθ. In dimension n= 3, we know that M# αβis the adjoint matrix of Mαβ. If we diagonalize Mαβwith eigenvalues λ≥µ≥νso that (Mαβ) = λ µ ν , thenM2 αβandM# αβare also diagonal, with (M2 αβ) = λ2 µ2 ν2 and (M# αβ) = µν λν λµ . Thus the ODE corresponding to PDE (2.4.1) for Mαβ(in the space of 3 ×3 matrices) is given by the following system (2.4.2)  d dtλ=λ2+µν, d dtµ=µ2+λν, d dtν=ν2+λµ. 224 H.-D. CAO AND X.-P. ZHU LetPbe the principal bundle of the manifold and form the associat ed bundle V=P×GE, whereG=O(3) andEis the vector space of symmetric bilinear forms onso(3). The curvature operator Mαβis a smooth section of V=P×GE. According to Theorem 2.3.1, any closed convex set of curvature operato r matricesMαβwhich isO(3)-invariant (and hence invariant under parallel transla tion) and preserved by ODE (2.4.2) is also preserved by the Ricci ﬂow. We are now ready to state and prove the Hamilton-Ivey pinching estimate . Theorem 2.4.1 ( Hamilton [63], Ivey [73] ).Suppose we have a solution to the Ricci ﬂow on a three-manifold which is complete with bounded curvature for each t≥0. Assume at t= 0the eigenvalues λ≥µ≥νof the curvature operator at each point are bounded below by ν≥ −1. The scalar curvature R=λ+µ+νis their sum. Then at all points and all times t≥0we have the pinching estimate R≥(−ν)[log(−ν)−3], wheneverν <0. Proof. Consider the function y=f(x) =x(logx−3) deﬁned on e2≤x <+∞. It is easy to check that fis increasing and convex with range−e2≤y<+∞. Letf−1(y) =xbe the inverse function, which is also increasing but concave and satisﬁes (2.4.3) lim y→∞f−1(y) y= 0 Consider also the set Kof matrices Mαβdeﬁned by the inequalities (2.4.4) K:  λ+µ+ν≥ −3, ν+f−1(λ+µ+ν)≥0. By Theorem 2.3.1 and the assumptions in Theorem 2.4.1 at t= 0, we only need to check that the set Kdeﬁned above is closed, convex and preserved by the ODE (2.4.2). ClearlyKis closed because f−1is continuous. λ+µ+νis just the trace function of 3×3 matrices which is a linear function. Hence the ﬁrst inequal ity in (2.4.4) deﬁnes a linear half-space, which is convex. The function νis the least eigenvalue function, which is concave. Also note that f−1is concave. Thus the second inequality in (2.4.4) deﬁnes a convex set as well. Therefore we proved Kis closed and convex. Under the ODE (2.4.2) d dt(λ+µ+ν) =λ2+µ2+ν2+λµ+λν+µν =1 2[(λ+µ)2+ (λ+ν)2+ (µ+ν)2] ≥0. Thus the ﬁrst inequality in (2.4.4) is preserved by the ODE. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 225 The second inequality in (2.4.4) can be written as λ+µ+ν≥f(−ν),wheneverν≤ −e2, which becomes (2.4.5) λ+µ≥(−ν)[log(−ν)−2],wheneverν≤ −e2. To show the inequality is preserved we only need to look at poi nts on the boundary of the set. Ifν+f−1(λ+µ+ν) = 0 thenν=−f−1(λ+µ+ν)≤ −e2sincef−1(y)≥e2. Hence the RHS of (2.4.5) is nonnegative. We thus have λ≥0 becauseλ≥µ. Butµ may have either sign. We split our consideration into two cas es: Case(i):µ≥0. We need to verify dλ dt+dµ dt≥(log(−ν)−1)d(−ν) dt whenλ+µ= (−ν)[log(−ν)−2]. Solving for log(−ν)−2 =λ+µ (−ν) and substituting above, we must show λ2+µν+µ2+λν≥/parenleftbiggλ+µ (−ν)+ 1/parenrightbigg (−ν2−λµ) which is equivalent to (λ2+µ2)(−ν) +λµ(λ+µ+ (−ν)) + (−ν)3≥0. Sinceλ,µand (−ν) are all nonnegative we are done in the ﬁrst case. Case(ii):µ<0. We need to verify dλ dt≥d(−µ) dt+ (log( −ν)−1)d(−ν) dt whenλ= (−µ) + (−ν)[log(−ν)−2]. Solving for log(−ν)−2 =λ−(−µ) (−ν) and substituting above, we need to show λ2+µν≥ −µ2−λν+/parenleftbiggλ−(−µ) (−ν)+ 1/parenrightbigg (−ν2−λµ) or λ2+ (−µ)(−ν)≥λ(−ν)−(−µ)2+/parenleftbiggλ−(−µ) (−ν)+ 1/parenrightbigg (λ(−µ)−(−ν)2) 226 H.-D. CAO AND X.-P. ZHU which reduces to λ2(−ν) +λ(−µ)2+ (−µ)2(−ν) + (−ν)3≥λ2(−µ) +λ(−µ)(−ν) or equivalently (λ2−λ(−µ) + (−µ)2)((−ν)−(−µ)) + (−µ)3+ (−ν)3≥0. Sinceλ2−λ(−µ) + (−µ)2≥0 and ( −ν)−(−µ)≥0 we are also done in the second case. Therefore the proof is completed. 2.5. Li-Yau-Hamilton Estimates. In [82], Li-Yau developed a fundamental gradient estimate, now called Li-Yau estimate, for positiv e solutions to the heat equa- tion on a complete Riemannian manifold with nonnegative Ric ci curvature. They used it to derive the Harnack inequality for such solutions b y path integration. Then based on the suggestion of Yau, Hamilton [60] developed a sim ilar estimate for the scalar curvature of solutions to the Ricci ﬂow on a Riemann su rface with positive curvature, and later obtained a matrix version of the Li-Yau estimate for solutions to the Ricci ﬂow with positive curvature operator in all dimens ions. This matrix version of the Li-Yau estimate is the Li-Yau-Hamilton estimate , which we will present in this section. The Li-Yau-Hamilton estimate plays a centr al role in the analysis of formation of singularities and the application of the Ric ci ﬂow to three-manifold topology. We have seen that in the Ricci ﬂow the curvature tensor satisﬁ es a nonlinear heat equation, and the nonnegativity of the curvature opera tor is preserved by the Ricci ﬂow. Roughly speaking the Li-Yau-Hamilton estimate s ays the nonnegativity of a certain combination of the derivatives of the curvature up to second order is also preserved by the Ricci ﬂow. Let us begin by describing the Li-Yau estimate for positive s olutions to the heat equation on a complete Riemannian manifold with nonnegativ e Ricci curvature. Theorem 2.5.1 ( Li-Yau [82] ).Let(M,g ij)be ann-dimensional complete Rie- mannian manifold with nonnegative Ricci curvature. Let u(x,t)be any positive solu- tion to the heat equation ∂u ∂t= ∆uonM×[0,∞). Then we have (2.5.1)∂u ∂t−|∇u|2 u+n 2tu≥0onM×(0,∞). We remark that one can in fact prove the following quadratic v ersion that for any vector ﬁeld VionM, (2.5.2)∂u ∂t+ 2∇u·V+u|V|2+n 2tu≥0. If we take the optimal vector ﬁeld V=−∇u/u, we recover the inequality (2.5.1). Now we consider the Ricci ﬂow on a Riemann surface. Since in di mension two the Ricci curvature is given by Rij=1 2Rgij, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 227 the Ricci ﬂow (1.1.5) becomes (2.5.3)∂gij ∂t=−Rgij. Now letgij(x,t) be a complete solution of the Ricci ﬂow (2.5.3) on a Riemann surfaceMand 0 ≤t<T. Then the scalar curvature R(x,t) evolves by the semilinear equation ∂R ∂t=△R+R2 onM×[0,T). Suppose the scalar curvature of the initial metric is boun ded, nonneg- ative everywhere and positive somewhere. Then it follows fr om Proposition 2.1.2 that the scalar curvature R(x,t) of the evolving metric remains nonnegative. Moreover, from the standard strong maximum principle (which works in e ach local coordinate neighborhood), the scalar curvature is positive everywher e fort>0. In [60], Hamilton obtained the following Li-Yau estimate for the scalar curva tureR(x,t). Theorem 2.5.2 ( Hamilton [60] ).Letgij(x,t)be a complete solution of the Ricci ﬂow on a surface M. Assume the scalar curvature of the initial metric is bounde d, nonnegative everywhere and positive somewhere. Then the sc alar curvature R(x,t) satisﬁes the Li-Yau estimate (2.5.4)∂R ∂t−|∇R|2 R+R t≥0. Proof. By the above discussion, we know R(x,t)>0 fort>0. If we set L= logR(x,t) fort>0, then ∂ ∂tL=1 R(△R+R2) =△L+|∇L|2+R and (2.5.4) is equivalent to ∂L ∂t− |∇L|2+1 t=△L+R+1 t≥0. Following Li-Yau [82] in the linear heat equation case, we co nsider the quantity (2.5.5) Q=∂L ∂t− |∇L|2=△L+R. Then by a direct computation, ∂Q ∂t=∂ ∂t(△L+R) =△/parenleftbigg∂L ∂t/parenrightbigg +R△L+∂R ∂t =△Q+ 2∇L· ∇Q+ 2|∇2L|2+ 2R(△L) +R2 ≥ △Q+ 2∇L· ∇Q+Q2. 228 H.-D. CAO AND X.-P. ZHU So we get ∂ ∂t/parenleftbigg Q+1 t/parenrightbigg ≥ △/parenleftbigg Q+1 t/parenrightbigg + 2∇L· ∇/parenleftbigg Q+1 t/parenrightbigg +/parenleftbigg Q−1 t/parenrightbigg/parenleftbigg Q+1 t/parenrightbigg . Hence by a similar maximum principle argument as in the proof of Lemma 2.1.3, we obtain Q+1 t≥0. This proves the theorem. As an immediate consequence, we obtain the following Harnac k inequality for the scalar curvature Rby taking the Li-Yau type path integral as in [82]. Corollary 2.5.3 ( Hamilton [60] ).Letgij(x,t)be a complete solution of the Ricci ﬂow on a surface with bounded and nonnegative scalar cu rvature. Then for any pointsx1,x2∈M, and 0<t1<t2, we have R(x2,t2)≥t1 t2e−dt1(x1,x2)2/4(t2−t1)R(x1,t1). Proof. Take the geodesic path γ(τ),τ∈[t1,t2],fromx1tox2at timet1with constant velocity dt1(x1,x2)/(t2−t1).Consider the space-time path η(τ) = (γ(τ),τ), τ∈[t1,t2]. We compute logR(x2,t2) R(x1,t1)=/integraldisplayt2 t1d dτL(γ(τ),τ)dτ =/integraldisplayt2 t11 R/parenleftbigg∂R ∂τ+∇R·dγ dτ/parenrightbigg dτ ≥/integraldisplayt2 t1/parenleftigg ∂L ∂τ− |∇L|2 gij(τ)−1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingledγ dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 gij(τ)/parenrightigg dτ. Then by Theorem 2.5.2 and the fact that the metric is shrinkin g (since the scalar curvature is nonnegative), we have logR(x2,t2) R(x1,t1)≥/integraldisplayt2 t1/parenleftigg −1 τ−1 4/vextendsingle/vextendsingle/vextendsingle/vextendsingledγ dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 gij(τ)/parenrightigg dτ = logt1 t2−dt1(x1,x2)2 4(t2−t1) After exponentiating above, we obtain the desired Harnack i nequality. To prove a similar inequality as (2.5.4) for the scalar curva ture of solutions to the Ricci ﬂow in higher dimensions is not so simple. First of all, we will need to require nonnegativity of the curvature operator (which we know is pr eserved under the Ricci ﬂow). Secondly, one does not get inequality (2.5.4) directl y, but rather indirectly as the trace of certain matrix estimate. The key ingredient in f ormulating this matrix version is to derive some identities from the soliton soluti ons and prove an elliptic THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 229 inequality based on these quantities. Hamilton found such a general principle which was based on the idea of Li-Yau [82] when an identity is checke d on the heat kernel before an inequality was found. To illustrate this point, le t us ﬁrst examine the heat equation case. Consider the heat kernel u(x,t) = (4πt)−n/2e−|x|2/4t for the standard heat equation on Rnwhich can be considered as an expanding soliton solution. Diﬀerentiating the function u, we get (2.5.6) ∇ju=−uxj 2tor∇ju+uVj= 0, where Vj=xj 2t=−∇ju u. Diﬀerentiating (2.5.6), we have (2.5.7) ∇i∇ju+∇iuVj+u 2tδij= 0. To make the expression in (2.5.7) symmetric in i,j, we multiply Vito (2.5.6) and add to (2.5.7) and obtain (2.5.8) ∇i∇ju+∇iuVj+∇juVi+uViVj+u 2tδij= 0. Taking the trace in (2.5.8) and using the equation ∂u/∂t = ∆u, we arrive at ∂u ∂t+ 2∇u·V+u|V|2+n 2tu= 0, which shows that the Li-Yau inequality (2.5.1) becomes an eq uality on our expanding soliton solution u! Moreover, we even have the matrix identity (2.5.8). Based on the above observation and using a similar process, H amilton found a matrix quantity, which vanishes on expanding gradient Ricc i solitons and is nonneg- ative for any solution to the Ricci ﬂow with nonnegative curv ature operator. Now we describe the process of ﬁnding the Li-Yau-Hamilton quadrat ic for the Ricci ﬂow in arbitrary dimension. Consider a homothetically expanding gradient soliton g, we have (2.5.9) Rab+1 2tgab=∇aVb in the orthonormal frame coordinate chosen as in Section 1.3 . HereVb=∇bffor some function f. Diﬀerentiating (2.5.9) and commuting give the ﬁrst order r elations ∇aRbc− ∇ bRac=∇a∇bVc− ∇ b∇aVc (2.5.10) =RabcdVd, and diﬀerentiating again, we get ∇a∇bRcd− ∇ a∇cRbd=∇a(RbcdeVe) =∇aRbcdeVe+Rbcde∇aVe =∇aRbcdeVe+RaeRbcde+1 2tRbcda. 230 H.-D. CAO AND X.-P. ZHU We further take the trace of this on aandbto get ∆Rcd− ∇ a∇cRad−RaeRacde+1 2tRcd− ∇ aRacdeVe= 0, and then by commuting the derivatives and second Bianchi ide ntity, ∆Rcd−1 2∇c∇dR+ 2RcadeRae−RceRde+1 2tRcd+ (∇eRcd− ∇ dRce)Ve= 0. Let us deﬁne Mab= ∆Rab−1 2∇a∇bR+ 2RacbdRcd−RacRbc+1 2tRab, Pabc=∇aRbc− ∇ bRac. Then (2.5.11) Mab+PcbaVc= 0, We rewrite (2.5.10) as Pabc=RabcdVd and then (2.5.12) PcabVc+RacbdVcVd= 0. Adding (2.5.11) and (2.5.12) we have Mab+ (Pcab+Pcba)Vc+RacbdVcVd= 0 and then MabWaWb+ (Pcab+Pcba)WaWbVc+RacbdWaVcWbVd= 0. If we write Uab=1 2(VaWb−VbWa) =V∧W, then the above identity can be rearranged as (2.5.13) Q∆=MabWaWb+ 2PabcUabWc+RabcdUabUcd= 0. This is the Li-Yau-Hamilton quadratic we look for. Note that the proof of the Li- Yau-Hamilton estimate below does not depend on the existenc e of such an expanding gradient Ricci soliton. It is only used as inspiration. Now we are ready to state the remarkable Li-Yau-Hamilton estimate for the Ricci ﬂow. Theorem 2.5.4 ( Hamilton [61] ).Letgij(x,t)be a complete solution with bounded curvature to the Ricci ﬂow on a manifold Mfortin some time interval (0,T)and suppose the curvature operator of gij(x,t)is nonnegative. Then for any one-form Wa and any two-form Uabwe have MabWaWb+ 2PabcUabWc+RabcdUabUcd≥0 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 231 onM×(0,T). The proof of this theorem requires some rather intense calcu lations. Here we only give a sketch of the proof. For more details, we refer the read er to Hamilton’s original paper [61]. Sketch of the Proof. Letgij(x,t) be the complete solution with bounded and nonnegative curvature operator. Recall that in the orthono rmal frame coordinate system, the curvatures evolve by   ∂ ∂tRabcd= ∆Rabcd+ 2(Babcd−Babdc−Badbc+Bacbd), ∂ ∂tRab= ∆Rab+ 2RacbdRcd, ∂ ∂tR= ∆R+ 2|Ric|2, whereBabcd=RaebfRced f. By a long but straightforward computation from these evolut ion equations, one can get /parenleftbigg∂ ∂t−∆/parenrightbigg Pabc= 2RadbePdec+ 2RadcePdbe+ 2RbdcePade−2Rde∇dRabce and /parenleftbigg∂ ∂t−∆/parenrightbigg Mab= 2RacbdMcd+ 2Rcd(∇cPdab+∇cPdba) + 2PacdPbcd−4PacdPbdc+ 2RcdRceRadbe−1 2t2Rab. Now consider Q∆=MabWaWb+ 2PabcUabWc+RabcdUabUcd. At a point where (2.5.14)/parenleftbigg∂ ∂t−∆/parenrightbigg Wa=1 tWa,/parenleftbigg∂ ∂t−∆/parenrightbigg Uab= 0, and (2.5.15) ∇aWb= 0,∇aUbc=1 2(RabWc−RacWb) +1 4t(gabWc−gacWb), we have /parenleftbigg∂ ∂t−∆/parenrightbigg Q= 2RacbdMcdWaWb−2PacdPbdcWaWb (2.5.16) + 8RadcePdbeUabWc+ 4RaecfRbed fUabUcd + (PabcWc+RabcdUcd)(PabeWe+RabefUef). For simplicity we assume the manifold is compact and the curv ature operator is strictly positive. (For the general case we shall mess the fo rmula up a bit to sneak in the termǫeAtf, as done in Lemma 2.1.3). Suppose not; then there will be a ﬁrs t time 232 H.-D. CAO AND X.-P. ZHU when the quantity Qis zero, and a point where this happens, and a choice of Uand Wgiving the null eigenvectors. We can extend UandWany way we like in space and time and still have Q≥0, up to the critical time. In particular we can make the ﬁrst derivatives in space and time to be anything we like, so w e can extend ﬁrst in space to make (2.5.15) hold at that point. And then, knowing ∆ Waand ∆Uab, we can extend in time to make (2.5.14) hold at that point and that moment. Thus we have (2.5.16) at the point. In the RHS of (2.5.16) the quadratic term (PabcWc+RabcdUcd)(PabeWe+RabefUef) is clearly nonnegative. By similar argument as in the proof o f Lemma 2.1.3, to get a contradiction we only need to show the remaining part in the R HS of (2.5.16) is also nonnegative. A nonnegative quadratic form can always be written as a sum of squares of lin- ear forms. This is equivalent to diagonalizing a symmetric m atrix and writing each nonnegative eigenvalue as a square. Write Q=/summationdisplay k(Xk aWa+Yk abUab)2,/parenleftbigg 1≤k≤n+n(n−1) 2/parenrightbigg . This makes Mab=/summationdisplay kXk aXk b, P abc=/summationdisplay kYk abXk c and Rabcd=/summationdisplay kYk abYk cd. It is then easy to compute 2RacbdMcdWaWb−2PacdPbdcWaWb+ 8RadcePdbeUabWe + 4RaecfRbed fUabUcd = 2/parenleftigg/summationdisplay kYk acYk bd/parenrightigg/parenleftigg/summationdisplay lXl aYl c/parenrightigg WaWb −2/parenleftigg/summationdisplay kYk acXk d/parenrightigg/parenleftigg/summationdisplay lYl bdXl c/parenrightigg WaWb + 8/parenleftigg/summationdisplay kYk adYk ce/parenrightigg/parenleftigg/summationdisplay lYl dbXl e/parenrightigg UabWc + 4/parenleftigg/summationdisplay kYk aeYk cf/parenrightigg/parenleftigg/summationdisplay lYl beYl d f/parenrightigg UabUcd =/summationdisplay k,l(Yk acXl cWa−Yl acXk cWa−2Yk acYl bcUab)2 ≥0. This says that the remaining part in the RHS of (2.5.16) is als o nonnegative. There- fore we have completed the sketch of the proof. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 233 By takingUab=1 2(VaWb−VbWa) and tracing over Wa, we immediately get Corollary 2.5.5 ( Hamilton [61] ).For any one-form Vawe have ∂R ∂t+R t+ 2∇aR·Va+ 2RabVaVb≥0. In particular by taking V≡0, we see that the function tR(x,t) is pointwise nondecreasing in time. By combining this property with the l ocal derivative estimate of curvature, we have the following elliptic type estimate. Corollary 2.5.6. Suppose we have a solution to the Ricci ﬂow for t>0which is complete with bounded curvature, and has nonnegative cur vature operator. Suppose also that at some time t>0we have the scalar curvature R≤Mfor some constant M in the ball of radius raround some point p. Then for k= 1,2,..., thekthorder derivatives of the curvature at pat the time tsatisfy a bound |∇kRm(p,t)|2≤CkM2/parenleftbigg1 r2k+1 tk+Mk/parenrightbigg for some constant Ckdepending only on the dimension and k. Proof. SincetRis nondecreasing in time, we get a bound R≤2Min the given region for times between t/2 andt. The nonnegative curvature hypothesis tells us the metric is shrinking. So we can apply the local derivative est imate in Theorem 1.4.2 to deduce the result. By a similar argument as in Corollary 2.5.3, one readily has t he following Harnack inequality. Corollary 2.5.7. Letgij(x,t)be a complete solution of the Ricci ﬂow on a manifold with bounded and nonnegative curvature operator, and letx1,x2∈M,0< t1<t2.Then the following inequality holds R(x2,t2)≥t1 t2e−dt1(x1,x2)2/2(t2−t1)·R(x1,t1). In the above discussion, we assumed that the solution to the R icci ﬂow exists on 0≤t<T, and we derived the Li-Yau-Hamilton estimate with terms 1 /tin it. When the solution happens to be ancient , i.e., deﬁned on −∞< t < T , Hamilton [61] found an interesting and simple procedure for getting rid of them. Suppose we have a solution on α<t<T we can replace tbyt−αin the Li-Yau-Hamilton estimate. If we letα→ −∞ , then the expression 1 /(t−α)→0 and disappears! In particular the trace Li-Yau-Hamilton estimate in Corollary 2.5.5 beco mes (2.5.17)∂R ∂t+ 2∇aR·Va+ 2RabVaVb≥0. By takingV= 0, we see that∂R ∂t≥0. Thus, we have the following Corollary 2.5.8 ( Hamilton [61] ).Letgij(x,t)be a complete ancient solution of the Ricci ﬂow on M×(−∞,T)with bounded and nonnegative curvature operator, then the scalar curvature R(x,t)is pointwise nondecreasing in time t. 234 H.-D. CAO AND X.-P. ZHU Corollary 2.5.8 will be very useful later on when we study anc ientκ-solutions in Chapter 6, especially combined with Shi’s derivative estim ate. We end this section by stating the Li-Yau-Hamilton estimate for the K¨ ahler-Ricci ﬂow, due to the ﬁrst author [12], under the weaker curvature a ssumption of nonneg- ative holomorphic bisectional curvature. Note that the fol lowing Li-Yau-Hamilton estimate in the K¨ ahler case is really a Li-Yau-Hamilton est imate for the Ricci tensor of the evolving metric, so not only can we derive an estimate o n the scalar curva- ture, which is the trace of the Ricci curvature, similar to Co rollary 2.5.5 but also an estimate on the determinant of the Ricci curvature as well. Theorem 2.5.9 ( Cao [12] ).Letgα¯β(x,t)be a complete solution to the K¨ ahler- Ricci ﬂow on a complex manifold Mwith bounded curvature and nonnegtive bisectional curvature and 0≤t<T. For any point x∈Mand any vector Vin the holomorphic tangent space T1,0 xM, let Qα¯β=∂ ∂tRα¯β+Rα¯γRγ¯β+∇γRα¯βVγ+∇¯γRα¯βV¯γ+Rα¯βγ¯δVγV¯δ+1 tRα¯β. Then we have Qα¯βWαW¯β≥0 for allx∈M,V,W∈T1,0 xM, andt>0. Corollary 2.5.10 ( Cao [12] ).Under the assumptions of Theorem 2.5.9,we have (i) the scalar curvature Rsatisﬁes the estimate ∂R ∂t−|∇R|2 R+R t≥0, and (ii) assuming Rα¯β>0, the determinant φ= det(Rα¯β)/det(gα¯β)of the Ricci curvature satisﬁes the estimate ∂φ ∂t−|∇φ|2 nφ+nφ t≥0 for allx∈Mandt>0. 2.6. Perelman’s Estimate for Conjugate Heat Equations. In [103] Perel- man obtained a Li-Yau type estimate for fundamental solutio ns of the conjugate heat equation, which is a backward heat equation, when the metric evolves by the Ricci ﬂow. In this section we shall describe how to get this estimat e along the same line as in the previous section. More importantly, we shall show how th e Li-Yau path integral, when applied to Perelman’s Li-Yau type estimate, leads to an important space-time distance function introduced by Perelman [103]. We learned from Hamilton [67] this idea of looking at Perelman’s Li-Yau estimate. We saw in the previous section that the Li-Yau quantity and th e Li-Yau-Hamilton quantity vanish on expanding solutions. Note that when we co nsider a backward heat equation, shrinking solitons can be viewed as expanding bac kward in time. So we start by looking at shrinking gradient Ricci solitons. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 235 Suppose we have a shrinking gradient Ricci soliton gijwith potential function f on manifold Mand−∞<t< 0 so that, for τ=−t, (2.6.1) Rij+∇j∇if−1 2τgij= 0. Then, by taking the trace, we have (2.6.2) R+ ∆f−n 2τ= 0. Also, by similar calculations as in deriving (1.1.15), we ge t (2.6.3) R+|∇f|2−f τ=C whereCis a constant which we can set to be zero. Moreover, observe (2.6.4)∂f ∂t=|∇f|2 becausefevolves in time with the rate of change given by the Lie deriva tive in the direction of ∇fgenerating the one-parameter family of diﬀeomorphisms. Combining (2.6.2) with (2.6.4), we see fsatisﬁes the backward heat equation (2.6.5)∂f ∂t=−∆f+|∇f|2−R+n 2τ, or equivalently (2.6.6)∂f ∂τ= ∆f− |∇f|2+R−n 2τ. Recall the Li-Yau-Hamilton quadratic is a certain combinat ion of the second order space derivative (or ﬁrst order time derivative), ﬁrst orde r space derivatives and zero orders. Multiplying (2.6.2) by a factor of 2 and subtracting (2.6.3) yields 2∆f− |∇f|2+R+1 τf−n τ= 0 valid for our potential function fof the shrinking gradient Ricci soliton. The quantity on the LHS of the above identity is precisely the Li-Yau-Hami lton type quadratic found by Perelman [103]. Note that a function fsatisﬁes the backward heat equation (2.6.6) if and only if the function u= (4πτ)−n 2e−f satisﬁes the so called conjugate heat equation (2.6.7) /square∗u/defines∂u ∂τ−∆u+Ru= 0. Lemma 2.6.1 ( Perelman [103] ).Letgij(x,t),0≤t<T, be a complete solution to the Ricci ﬂow on an n-dimensional manifold Mand letu= (4πτ)−n 2e−fbe a solution to the conjugate equation (2.6.7)withτ=T−t. Set H= 2∆f− |∇f|2+R+f−n τ 236 H.-D. CAO AND X.-P. ZHU and v=τHu=/parenleftbig τ(R+ 2∆f− |∇f|2) +f−n/parenrightbig u. Then we have ∂H ∂τ= ∆H−2∇f· ∇H−1 τH−2/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , and ∂v ∂τ= ∆v−Rv−2τu/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . Proof. By direct computations, we have ∂ ∂τH=∂ ∂τ/parenleftbigg 2△f− |∇f|2+R+f−n τ/parenrightbigg = 2△/parenleftbigg∂f ∂τ/parenrightbigg −2/a\}b∇acketle{t2Rij,fij/a\}b∇acket∇i}ht −2/angbracketleftbigg ∇f,∇/parenleftbigg∂f ∂τ/parenrightbigg/angbracketrightbigg + 2Ric ( ∇f,∇f) +∂ ∂τR+1 τ∂ ∂τf−f−n τ2 = 2△/parenleftig △f− |∇f|2+R−n 2τ/parenrightig −4/a\}b∇acketle{tRij,fij/a\}b∇acket∇i}ht+ 2Ric ( ∇f,∇f) −2/angbracketleftig ∇f,∇/parenleftig △f− |∇f|2+R−n 2τ/parenrightig/angbracketrightig − △R−2|Rij|2 +1 τ/parenleftig △f− |∇f|2+R−n 2τ/parenrightig −f−n τ2, ∇H=∇/parenleftbigg 2△f− |∇f|2+R+f−n τ/parenrightbigg = 2∇(△f)−2/a\}b∇acketle{t∇∇ if,∇if/a\}b∇acket∇i}ht+∇R+1 τ∇f, △H=△/parenleftbigg 2△f− |∇f|2+R+f−n τ/parenrightbigg = 2△(△f)− △(|∇f|2) +△R+1 τ△f, and 2∇H· ∇f= 2/a\}b∇acketle{t2∇(△f)−2/a\}b∇acketle{t∇∇ if,∇if/a\}b∇acket∇i}ht+∇R+1 τ∇f,∇f/a\}b∇acket∇i}ht = 2 [/a\}b∇acketle{t2∇(△f),∇f/a\}b∇acket∇i}ht −2/a\}b∇acketle{tfij,fifj/a\}b∇acket∇i}ht+/a\}b∇acketle{t∇R,∇f/a\}b∇acket∇i}ht] +2 τ|∇f|2. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 237 Thus we get ∂ ∂τH− △H+ 2∇f· ∇H+1 τH =−4/a\}b∇acketle{tRij,fij/a\}b∇acket∇i}ht+ 2Ric ( ∇f,∇f)−2|Rij|2− △(|∇f|2) + 2/a\}b∇acketle{t∇(△f),∇f/a\}b∇acket∇i}ht +2 τ△f+2 τR−n 2τ2 =−2/bracketleftbigg |Rij|2+|fij|2+n 4τ2+ 2/a\}b∇acketle{tRij,fij/a\}b∇acket∇i}ht −R τ−1 τ△f/bracketrightbigg =−2/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , and /parenleftbigg∂ ∂τ− △+R/parenrightbigg v=/parenleftbigg∂ ∂τ− △+R/parenrightbigg (τHu) =/parenleftbigg∂ ∂τ− △/parenrightbigg (τH)·u−2/a\}b∇acketle{t∇(τH),∇u/a\}b∇acket∇i}ht =/parenleftbigg/parenleftbigg∂ ∂τ− △/parenrightbigg (τH)−2/a\}b∇acketle{t∇(τH),∇f/a\}b∇acket∇i}ht/parenrightbigg u =τ/parenleftbigg∂H ∂τ− △H+ 2∇f· ∇H+1 τH/parenrightbigg u =−2τu/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . Note that, since fsatisﬁes the equation (2.6.6), we can rewrite Has (2.6.8) H= 2∂f ∂τ+|∇f|2−R+1 τf. Then, by Lemma 2.6.1, we have ∂ ∂τ(τH) = ∆(τH)−2∇f· ∇(τH)−2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingleRic + ∇2f−1 2τg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . So by the maximum principle, we ﬁnd max( τH) is nonincreasing as τincreasing. Whenuis chosen to be a fundamental solution to (2.6.7), one can sho w that limτ→0τH≤0 and hence H≤0 onMfor allτ∈(0,T] (see, for example, [99]). Since this fact is not used in later chapters and will be only u sed in the rest of the sec- tion to introduce a space-time distance via Li-Yau path inte gral, we omit the details of the proof. Once we have Perelman’s Li-Yau type estimate H≤0, we can apply the Li-Yau path integral as in [82] to estimate the above solution u(i.e., a heat kernel estimate for the conjugate heat equation, see also the earlier work of Cheeger-Yau [28]). Let p,q∈Mbe two points and γ(τ),τ∈[0,¯τ],be a curve joining pandq, withγ(0) =p 238 H.-D. CAO AND X.-P. ZHU andγ(¯τ) =q. Then along the space-time path ( γ(τ),τ),τ∈[0,¯τ], we have d dτ/parenleftbig 2√τf(γ(τ),τ)/parenrightbig = 2√τ/parenleftbigg∂f ∂τ+∇f·˙γ(τ)/parenrightbigg +1√τf ≤√τ/parenleftbig − |∇f|2 gij(τ)+ 2∇f·˙γ(τ)/parenrightbig +√τR =−√τ|∇f−˙γ(τ)|2 gij(τ)+√τ(R+|˙γ(τ)|2 gij(τ)) ≤√τ(R+|˙γ(τ)|2 gij(τ)) where we have used the fact that H≤0 and the expression for Hin (2.6.8). Integrating the above inequality from τ= 0 toτ= ¯τ, we obtain 2√ ¯τf(q,¯τ)≤/integraldisplay¯τ 0√τ(R+|˙γ(τ)|2 gij(τ))dτ, or f(q,¯τ)≤1 2√¯τL(γ), where (2.6.9) L(γ)/defines/integraldisplay¯τ 0√τ(R+|˙γ(τ)|2 gij(τ))dτ. Denote by (2.6.10) l(q,¯τ)/definesinf γ1 2√¯τL(γ), where theinfis taken over all space curves γ(τ),0≤τ≤¯τ, joiningpandq. The space-time distance function l(q,¯τ) obtained by the above Li-Yau path integral ar- gument is ﬁrst introduced by Perelman in [103] and is what Per elman calls reduced distance. Since Perelman pointed out in page 19 of [103] that “an even closer refer- ence is [82], where they use ‘length’, associated to a linear parabolic equation, which is pretty much the same as in our case”, it is natural to call l(q,¯τ) theLi-Yau- Perelman distance . See Chapter 3 for much more detailed discussions. Finally, we conclude this section by relating the quantity H(orv) and the W- functional of Perelman deﬁned in (1.5.9). Observe that vhappens to be the integrand of the W-functional, W(gij(t),f,τ) =/integraldisplay MvdV. Hence, when Mis compact, d dτW=/integraldisplay M/parenleftbigg∂ ∂τv+Rv/parenrightbigg dV =−2τ/integraldisplay M/vextendsingle/vextendsingle/vextendsingle/vextendsingleRic + ∇2f−1 2τg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 udV ≤0, or equivalently, d dtW(gij(t),f(t),τ(t)) =/integraldisplay M2τ/vextendsingle/vextendsingle/vextendsingle/vextendsingleRij+∇i∇jf−1 2τgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (4πτ)−n 2e−fdV, which is the same as stated in Proposition 1.5.8. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 239 3. Perelman’s Reduced Volume. In Section 1.5 we introduced the F- functional and the W-functional of Perelman and proved their monotonicity prop - erties under the Ricci ﬂow. In the last section of the previou s chapter we have deﬁned the Li-Yau-Perelman distance. The main purpose of this chap ter is to use the Li-Yau- Perelman distance to deﬁne the Perelman’s reduced volume, w hich was introduced by Perelman in [103], and prove the monotonicity property of th e reduced volume under the Ricci ﬂow. This new monotonicity formula is more useful f or local considerations, especially when we consider the formation of singularities in Chapter 6 and work on the Ricci ﬂow with surgery in Chapter 7. As ﬁrst applications we will present two no local collapsing theorems of Perelman [103] in this chapter . More applications can be found in Chapter 6 and 7. 3.1. Riemannian Formalism in Potentially Inﬁnite Dimensio ns.In Sec- tion 2.6, from an analytic view point, we saw how the Li-Yau pa th integral of Perel- man’s estimate for fundamental solutions to the conjugate h eat equation leads to the Li-Yau-Perelman distance. In this section we present, from a geometric view point, another motivation why one is lead to the consideration of th e Li-Yau-Perelman dis- tance function, as well as a reduced volume concept. Interes tingly enough, the Li- Yau-Hamilton quadratic introduced in Section 2.5 appears a gain in this geometric consideration. We consider the Ricci ﬂow ∂ ∂tgij=−2Rij on a manifold Mwhere we assume that gij(·,t) are complete and have uniformly bounded curvatures. Recall from Section 2.5 that the Li-Yau-Hamilton quadratic introduced in [61] is Q=MijWiWj+ 2PijkUijWk+RijklUijUkl where Mij= ∆Rij−1 2∇i∇jR+ 2RikjlRkl−RikRjk+1 2tRij, Pijk=∇iRjk− ∇ jRik andUijis any two-form and Wiis any 1-form. Here and throughout this chapter we do not always bother to raise indices; repeated indices is sh ort hand for contraction with respect to the metric. In [63], Hamilton predicted that the Li-Yau-Hamilton quadr atic is some sort of jet extension of positive curvature operator on some larger spa ce. Such an interpretation of the Li-Yau-Hamilton quadratic as a curvature operator on the spaceM×R+was found by Chow and Chu [38] where a potentially degenerate Rie mannian metric on M×R+was constructed. The degenerate Riemannian metric on M×R+is the limit of the following two-parameter family of Riemannian metric s gN,δ(x,t) =g(x,t) + (R(x,t) +N 2(t+δ))dt2 asNtends to inﬁnity and δtends to zero, where g(x,t) is the solution of the Ricci ﬂow onMandt∈R+. 240 H.-D. CAO AND X.-P. ZHU To avoid the degeneracy, Perelman [103] considers the manif old˜M=M×SN×R+ with the following metric: ˜gij=gij, ˜gαβ=τgαβ, ˜goo=N 2τ+R, ˜giα= ˜gio= ˜gαo= 0, wherei,jare coordinate indices on M;α, βare coordinate indices on SN; and the coordinate τonR+has indexo. Letτ=T−tfor some ﬁxed constant T. Thengij will evolve with τby the backward Ricci ﬂow∂ ∂τgij= 2Rij. The metric gαβonSNis a metric with constant sectional curvature1 2N. We remark that the metric ˜ gαβonSNis chosen so that the product metric (˜gij,˜gαβ) onM×SNevolves by the Ricci ﬂow, while the component ˜ goois just the scalar curvature of (˜ gij,˜gαβ). Thus the metric ˜ gdeﬁned on ˜M=M×SN×R+is exactly a “regularization” of Chow-Chu’s degenerate metri c onM×R+. Proposition 3.1.1. The components of the curvature tensor of the metric ˜g coincide (moduloN−1)with the components of the Li-Yau-Hamilton quadratic. Proof. By deﬁnition, the Christoﬀel symbols of the metric ˜ gare given by the following list: ˜Γk ij= Γk ij, ˜Γk iβ= 0 and ˜Γγ ij= 0, ˜Γk αβ= 0 and ˜Γγ iβ= 0, ˜Γk io=gklRliand ˜Γo ij=−˜gooRij, ˜Γk oo=−1 2gkl∂ ∂xlRand ˜Γo io=1 2˜goo∂ ∂xiR, ˜Γo iβ= 0,˜Γk oβ= 0 and ˜Γγ oj= 0, ˜Γγ αβ= Γγ αβ, ˜Γγ αo=1 2τδγ αand ˜Γo αβ=−1 2˜googαβ, ˜Γγ oo= 0 and ˜Γo oβ= 0, ˜Γo oo=1 2˜goo/parenleftbigg −N 2τ2+∂ ∂τR/parenrightbigg . Fix a point ( p,s,τ)∈M×SN×R+and choose normal coordinates around p∈M and normal coordinates around s∈SNsuch that Γk ij(p) = 0 and Γγ αβ(s) = 0 for all i,j,k andα,β,γ . We compute the curvature tensor ˜Rmof the metric ˜ gat the point THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 241 as follows: ˜Rijkl=Rijkl+˜Γk io˜Γo jl−˜Γk jo˜Γo il=Rijkl+O/parenleftbigg1 N/parenrightbigg , ˜Rijkδ= 0, ˜Rijγδ= 0 and ˜Riβkδ=˜Γk io˜Γo βδ−˜Γk βo˜Γo iδ=−1 2˜googβδgklRli=O/parenleftbigg1 N/parenrightbigg , ˜Riβγδ= 0, ˜Rijko=∂ ∂xiRjk−∂ ∂xjRik+˜Γk io˜Γo jo−˜Γk jo˜Γo io=Pijk+O/parenleftbigg1 N/parenrightbigg , ˜Rioko=−1 2∂2 ∂xi∂xkR−∂ ∂τ(Rilglk) +˜Γk io˜Γo oo−˜Γk oj˜Γj io−˜Γk oo˜Γo io =−1 2∇i∇kR−∂ ∂τRik+ 2RikRlk−1 2τRik−RijRjk+O/parenleftbigg1 N/parenrightbigg =Mik+O(1 N), ˜Rijγo= 0 and ˜Riγjo= 0, ˜Riβγo=−τ˜Γγ βo˜Γo io=O/parenleftbigg1 N/parenrightbigg and ˜Rioγδ= 0, ˜Rioγo= 0, ˜Rαβγo= 0, ˜Rαoγo=/parenleftbigg1 2τ2δγ α+˜Γγ αo˜Γo oo−˜Γγ oβ˜Γβ αo/parenrightbigg τ=O/parenleftbigg1 N/parenrightbigg , ˜Rαβγδ=O/parenleftbigg1 N/parenrightbigg . Thus the components of the curvature tensor of the metric ˜ gcoincide (modulo N−1) with the components of the Li-Yau-Hamilton quadratic. The following observation due to Perelman [103] gives an imp ortant motivation to deﬁne Perelman’s reduced volume. Corollary 3.1.2. All components of the Ricci tensor of ˜gare zero (modulo N−1). Proof. From the list of the components of the curvature tensor of ˜ ggiven above, 242 H.-D. CAO AND X.-P. ZHU we have ˜Rij= ˜gkl˜Rijkl+ ˜gαβ˜Riαjβ+ ˜goo˜Riojo =Rij−1 2τgαβ˜googαβRij+ ˜goo/parenleftbigg Mij−1 2τRij+O/parenleftbigg1 N/parenrightbigg/parenrightbigg =Rij−N 2τ˜gooRij+O/parenleftbigg1 N/parenrightbigg =O/parenleftbigg1 N/parenrightbigg , ˜Riγ= ˜gkl˜Rikγl+ ˜gαβ˜Riαγβ+ ˜goo˜Rioγo= 0, ˜Rio= ˜gkl˜Rikol+ ˜gαβ˜Riαoβ+ ˜goo˜Riooo =−gklPikl+O/parenleftbigg1 N/parenrightbigg , ˜Rαβ= ˜gkl˜Rαkβl+ ˜gγδ˜Rαγβδ+ ˜goo˜Rαoβo =O/parenleftbigg1 N/parenrightbigg , ˜Rαo= ˜gkl˜Rαkol+ ˜gβγ˜Rαβoγ+ ˜goo˜Rαooo= 0, ˜Roo= ˜gkl˜Rokol+ ˜gαβ˜Roαoβ+ ˜goo˜Roooo =gkl/parenleftbigg Mkl+O/parenleftbigg1 N/parenrightbigg/parenrightbigg +O/parenleftbigg1 N/parenrightbigg . Since ˜goois of orderN−1, we see that the norm of the Ricci tensor is given by |˜Ric|˜g=O/parenleftbigg1 N/parenrightbigg . This proves the result. We now use the Ricci-ﬂatness of the metric ˜ gto interpret the Bishop-Gromov rel- ative volume comparison theorem which will motivate anothe r monotonicity formula for the Ricci ﬂow. The argument in the following will not be ri gorous. However it gives an intuitive picture of what one may expect. Consider a metric ball in ( ˜M,˜g) centered at some point ( p,s,0)∈˜M. Note that the metric of the sphere SNatτ= 0 degenerates and it shrinks to a point. Then the shortest geod esicγ(τ) between (p,s,0) and an arbitrary point ( q,¯s,¯τ)∈˜Mis always orthogonal to the SNﬁbre. The length ofγ(τ) can be computed as /integraldisplay¯τ 0/radicaligg/parenleftbiggN 2τ+R/parenrightbigg +|˙γ(τ)|2 gij(τ)dτ =√ 2N¯τ+1√ 2N/integraldisplay¯τ 0√τ(R+|˙γ(τ)|2 gij)dτ+O(N−3 2). Thus a shortest geodesic should minimize L(γ) =/integraldisplay¯τ 0√τ(R+|˙γ(τ)|2 gij)dτ. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 243 LetL(q,¯τ) denote the corresponding minimum. We claim that a metric sp here S˜M(√ 2N¯τ) in˜Mof radius√ 2N¯τcentered at ( p,s,0) isO(N−1)-close to the hyper- surface {τ= ¯τ}. Indeed, if ( x,s′,τ(x)) lies on the metric sphere S˜M(√ 2N¯τ), then the distance between ( x,s′,τ(x)) and (p,s,0) is √ 2N¯τ=/radicalbig 2Nτ(x) +1√ 2NL(x,τ(x)) +O/parenleftig N−3 2/parenrightig which can be written as /radicalbig τ(x)−√ ¯τ=−1 2NL(x,τ(x)) +O(N−2) =O(N−1). This shows that the metric sphere S˜M(√ 2N¯τ) isO(N−1)-close to the hypersurface {τ= ¯τ}. Note that the metric gαβonSNhas constant sectional curvature1 2N. Thus Vol/parenleftig S˜M/parenleftig√ 2N¯τ/parenrightig/parenrightig ≈/integraldisplay M/parenleftbigg/integraldisplay SNdVτ(x)gαβ/parenrightbigg dVgij(x) =/integraldisplay M(τ(x))N 2Vol (SN)dVM ≈(2N)N 2ωN/integraldisplay M/parenleftbigg√ ¯τ−1 2NL(x,τ(x)) +O(N−2)/parenrightbiggN dVM ≈(2N)N 2ωN/integraldisplay M/parenleftbigg√ ¯τ−1 2NL(x,¯τ) +o(N−1)/parenrightbiggN dVM, whereωNis the volume of the standard N-dimensional sphere. Now the volume of Euclidean sphere of radius√ 2N¯τinRn+N+1is Vol(SRn+N+1(√ 2N¯τ)) = (2N¯τ)N+n 2ωn+N. Thus we have Vol (S˜M(√ 2N¯τ)) Vol(SRn+N+1(√ 2N¯τ))≈const·N−n 2·/integraldisplay M(¯τ)−n 2exp/braceleftbigg −1 2√¯τL(x,¯τ)/bracerightbigg dVM. Since the Ricci curvature of ˜Mis zero (modulo N−1), the Bishop-Gromov volume comparison theorem then suggests that the integral ˜V(¯τ)∆=/integraldisplay M(4π¯τ)−n 2exp/braceleftbigg −1 2√¯τL(x,¯τ)/bracerightbigg dVM, which we will call Perelman’s reduced volume , should be nonincreasing in ¯ τ. A rigorous proof of this monotonicity property will be given i n the next section. One should note the analog of reduced volume with the heat kernel and there is a parallel calculation for the heat kernel of the Shr¨ odinger equation in the paper of Li-Yau [82]. 3.2. Comparison Theorems for Perelman’s Reduced Volume. In this section we will write the Ricci ﬂow in the backward version ∂ ∂τgij= 2Rij 244 H.-D. CAO AND X.-P. ZHU on a manifold Mwithτ=τ(t) satisfying dτ/dt =−1 (in practice we often take τ=t0−tfor some ﬁxed time t0). We always assume that either Mis compact or gij(τ) are complete and have uniformly bounded curvature. To each (smooth) space curveγ(τ), 0<τ1≤τ≤τ2, inM, we deﬁne its L-length as L(γ) =/integraldisplayτ2 τ1√τ(R(γ(τ),τ) +|˙γ(τ)|2 gij(τ))dτ. LetX(τ) = ˙γ(τ), and letY(τ) be any (smooth) vector ﬁeld along γ(τ). First of all, we compute the ﬁrst variation formula for L-length. Lemma 3.2.1 ( First variation formula ). δY(L) = 2√τ/a\}b∇acketle{tX,Y/a\}b∇acket∇i}ht|τ2τ1+/integraldisplayτ2 τ1√τ/angbracketleftbigg Y,∇R−2∇XX−4Ric (·,X)−1 τX/angbracketrightbigg dτ where /a\}b∇acketle{t·,·/a\}b∇acket∇i}htdenotes the inner product with respect to the metric gij(τ). Proof. By direct computations, δY(L) =/integraldisplayτ2 τ1√τ(/a\}b∇acketle{t∇R,Y/a\}b∇acket∇i}ht+ 2/a\}b∇acketle{tX,∇YX/a\}b∇acket∇i}ht)dτ =/integraldisplayτ2 τ1√τ(/a\}b∇acketle{t∇R,Y/a\}b∇acket∇i}ht+ 2/a\}b∇acketle{tX,∇XY/a\}b∇acket∇i}ht)dτ =/integraldisplayτ2 τ1√τ/parenleftbigg /a\}b∇acketle{t∇R,Y/a\}b∇acket∇i}ht+ 2d dτ/a\}b∇acketle{tX,Y/a\}b∇acket∇i}ht −2/a\}b∇acketle{t∇XX,Y/a\}b∇acket∇i}ht −4Ric(X,Y)/parenrightbigg dτ = 2√τ/a\}b∇acketle{tX,Y/a\}b∇acket∇i}ht|τ2 τ1+/integraldisplayτ2 τ1√τ/angbracketleftbigg Y,∇R−2∇XX−4Ric (·,X)−1 τX/angbracketrightbigg dτ. A smooth curve γ(τ) inMis called an L-geodesic if it satisﬁes the following L-geodesic equation (3.2.1) ∇XX−1 2∇R+1 2τX+ 2Ric (X,·) = 0. Given any two points p,q∈Mandτ2> τ1>0, there always exists an L-shortest curve (or shortest L-geodesic)γ(τ): [τ1,τ2]→Mconnecting ptoqwhich satisﬁes the above L-geodesic equation. Multiplying the L-geodesic equation (3.2.1) by√τ, we get ∇X(√τX) =√τ 2∇R−2√τRic (X,·) on [τ1,τ2], or equivalently d dτ(√τX) =√τ 2∇R−2Ric(√τX,·) on [τ1,τ2]. Thus if a continuous curve, deﬁned on [0 ,τ2], satisﬁes the L-geodesic equation on every subinterval 0 < τ1≤τ≤τ2, then√τ1X(τ1) has a limit as τ1→0+. This allows us to extend the deﬁnition of the L-length to include the case τ1= 0 for THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 245 all those (continuous) curves γ: [0,τ2]→Mwhich are smooth on (0 ,τ2] and have limits lim τ→0+√τ˙γ(τ). Clearly, there still exists an L-shortest curve γ(τ) : [0,τ2]→M connecting arbitrary two points p,q∈Mand satisfying the L-geodesic equation (3.2.1) on (0 ,τ2]. Moreover, for any vector v∈TpM, we can ﬁnd an L-geodesicγ(τ) starting at pwith lim τ→0+√τ˙γ(τ) =v. From now on, we ﬁx a point p∈Mand setτ1= 0. The L-distance function on the space-time M×R+is denoted by L(q,¯τ) and deﬁned to be the L-length of the L-shortest curve γ(τ) connecting pandqwith 0 ≤τ≤¯τ. Consider a shortest L-geodesicγ: [0,¯τ]→Mconnecting ptoq. In the computa- tions below we pretend that L-shortest geodesics between pandqare unique for all pairs (q,¯τ); if this is not the case, the inequalities that we obtain are still valid, by a standard barrier argument, when understood in the sense of distributions (see, for example, [112]). The ﬁrst variation formula in Lemma 3.2.1 implies that ∇YL(q,¯τ) =/angbracketleftig 2√ ¯τX(¯τ),Y(¯τ)/angbracketrightig . Thus ∇L(q,¯τ) = 2√ ¯τX(¯τ), and (3.2.2) |∇L|2= 4¯τ|X|2=−4¯τR+ 4¯τ(R+|X|2). We also compute L¯τ(γ(¯τ),¯τ) =d dτL(γ(τ),τ)|τ=¯τ− /a\}b∇acketle{t∇L,X/a\}b∇acket∇i}ht (3.2.3) =√ ¯τ(R+|X|2)−2√ ¯τ|X|2 = 2√ ¯τR−√ ¯τ(R+|X|2). To evaluate R+|X|2, we compute by using (3.2.1), d dτ(R(γ(τ),τ) +|X(τ)|2 gij(τ)) =Rτ+/a\}b∇acketle{t∇R,X/a\}b∇acket∇i}ht+ 2/a\}b∇acketle{t∇XX,X/a\}b∇acket∇i}ht+ 2Ric (X,X) =Rτ+1 τR+ 2/a\}b∇acketle{t∇R,X/a\}b∇acket∇i}ht −2Ric(X,X)−1 τ(R+|X|2) =−Q(X)−1 τ(R+|X|2), where Q(X) =−Rτ−R τ−2/a\}b∇acketle{t∇R,X/a\}b∇acket∇i}ht+ 2Ric(X,X) is the trace Li-Yau-Hamilton quadratic in Corollary 2.5.5. Hence d dτ(τ3 2(R+|X|2))|τ=¯τ=1 2√ ¯τ(R+|X|2)−¯τ3 2Q(X) =1 2d dτL(γ(τ),τ)|τ=¯τ−¯τ3 2Q(X). 246 H.-D. CAO AND X.-P. ZHU Therefore, (3.2.4) ¯ τ3 2(R+|X|2) =1 2L(q,¯τ)−K, where (3.2.5) K=/integraldisplay¯τ 0τ3 2Q(X)dτ. Combining (3.2.2) with (3.2.3), we obtain (3.2.6) |∇L|2=−4¯τR+2√¯τL−4√¯τK and (3.2.7) L¯τ= 2√ ¯τR−1 2¯τL+1 ¯τK. Next we compute the second variation of an L-geodesic. Lemma 3.2.2 ( Second variation formula ).For any L-geodesicγ, we have δ2 Y(L) = 2√τ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht|¯τ 0+/integraldisplay¯τ 0√τ[2|∇XY|2+ 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht +∇Y∇YR+ 2∇XRic (Y,Y)−4∇YRic (Y,X)]dτ. Proof. We compute δ2 Y(L) =Y/parenleftbigg/integraldisplay¯τ 0√τ(Y(R) + 2/a\}b∇acketle{t∇YX,X/a\}b∇acket∇i}ht)dτ/parenrightbigg =/integraldisplay¯τ 0√τ(Y(Y(R)) + 2/a\}b∇acketle{t∇Y∇YX,X/a\}b∇acket∇i}ht+ 2|∇YX|2)dτ =/integraldisplay¯τ 0√τ(Y(Y(R)) + 2/a\}b∇acketle{t∇Y∇XY,X/a\}b∇acket∇i}ht+ 2|∇XY|2)dτ and 2/a\}b∇acketle{t∇Y∇XY,X/a\}b∇acket∇i}ht = 2/a\}b∇acketle{t∇X∇YY,X/a\}b∇acket∇i}ht+ 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht = 2d dτ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht −4Ric(∇YY,X)−2/a\}b∇acketle{t∇YY,∇XX/a\}b∇acket∇i}ht −/parenleftbigg 2/angbracketleftbiggd dτ∇YY,X/angbracketrightbigg −2/a\}b∇acketle{t∇X∇YY,X/a\}b∇acket∇i}ht/parenrightbigg + 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht = 2d dτ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht −4Ric(∇YY,X)−2/a\}b∇acketle{t∇YY,∇XX/a\}b∇acket∇i}ht −2/angbracketleftbigg YiYj(gkl(∇iRlj+∇jRli− ∇ lRij))∂ ∂xk,X/angbracketrightbigg + 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht = 2d dτ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht −4Ric(∇YY,X)−2/a\}b∇acketle{t∇YY,∇XX/a\}b∇acket∇i}ht −4∇YRic (X,Y) + 2∇XRic(Y,Y) + 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 247 where we have used the computation ∂ ∂τΓk ij=gkl(∇iRlj+∇jRli− ∇ lRij). Thus by using the L-geodesic equation (3.2.1), we get δ2 Y(L) =/integraldisplay¯τ 0√τ/bracketleftbigg Y(Y(R)) + 2d dτ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht −4Ric (∇YY,X) −2/a\}b∇acketle{t∇YY,∇XX/a\}b∇acket∇i}ht −4∇YRic (X,Y) + 2∇XRic (Y,Y) + 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht+ 2|∇XY|2/bracketrightbigg dτ =/integraldisplay¯τ 0/bracketleftbigg 2√τd dτ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht+1√τ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht/bracketrightbigg dτ +/integraldisplay¯τ 0√τ[Y(Y(R))− /a\}b∇acketle{t∇ YY,∇R/a\}b∇acket∇i}ht −4∇YRic (X,Y) + 2∇XRic (Y,Y) + 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht+ 2|∇XY|2]dτ = 2√τ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht|¯τ 0+/integraldisplay¯τ 0√τ[2|∇XY|2+ 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht +∇Y∇YR−4∇YRic (X,Y) + 2∇XRic (Y,Y)]dτ. We now use the above second variation formula to estimate the Hessian of the L-distance function. Letγ(τ) : [0,¯τ]→Mbe an L-shortest curve connecting pandqso that the L-distance function L=L(q,¯τ) is given by the L-length ofγ. We ﬁx a vector Yat τ= ¯τwith|Y|gij(¯τ)= 1, and extend Yalong the L-shortest geodesic γon [0,¯τ] by solving the following ODE (3.2.8) ∇XY=−Ric(Y,·) +1 2τY. This is similar to the usual parallel translation and multip lication with proportional parameter. Indeed, suppose {Y1,...,Y n}is an orthonormal basis at τ= ¯τ(with respect to the metric gij(¯τ)) and extend this basis along the L-shortest geodesic γby solving the above ODE (3.2.8). Then d dτ/a\}b∇acketle{tYi,Yj/a\}b∇acket∇i}ht= 2Ric(Yi,Yj) +/a\}b∇acketle{t∇XYi,Yj/a\}b∇acket∇i}ht+/a\}b∇acketle{tYi,∇XYj/a\}b∇acket∇i}ht =1 τ/a\}b∇acketle{tYi,Yj/a\}b∇acket∇i}ht for alli,j. Hence, (3.2.9) /a\}b∇acketle{tYi(τ),Yj(τ)/a\}b∇acket∇i}ht=τ ¯τδij and{Y1(τ),...,Y n(τ)}remains orthogonal on [0 ,¯τ] withYi(0) = 0, i= 1,...,n . Proposition 3.2.3. Given any unit vector Yat any point q∈Mwithτ= ¯τ, consider an L-shortest geodesic γconnecting ptoqand extend Yalongγby solving 248 H.-D. CAO AND X.-P. ZHU theODE (3.2.8). Then the Hessian of the L-distance function LonMwithτ= ¯τ satisﬁes Hess L(Y,Y)≤1√¯τ−2√ ¯τRic (Y,Y)−/integraldisplay¯τ 0√τQ(X,Y)dτ in the sense of distributions, where Q(X,Y) =−∇Y∇YR−2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht −4∇XRic (Y,Y) + 4∇YRic (Y,X) −2Ric τ(Y,Y) + 2|Ric(Y,·)|2−1 τRic (Y,Y) is the Li-Yau-Hamilton quadratic. Moreover the equality ho lds if and only if the vector ﬁeldY(τ),τ∈[0,¯τ], is an L-Jacobian ﬁeld (i.e.,Yis the derivative of a variation ofγbyL-geodesics ). Proof. As said before, we pretend that the shortest L-geodesics between pand qare unique so that L(q,¯τ) is smooth. Otherwise, the inequality is still valid, by a standard barrier argument, when understood in the sense of d istributions (see, for example, [112]). Recall that ∇L(q,¯τ) = 2√¯τX. Then /a\}b∇acketle{t∇YY,∇L/a\}b∇acket∇i}ht= 2√¯τ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht.We compute by using Lemma 3.2.2, (3.2.8) and (3.2.9), Hess L(Y,Y) =Y(Y(L))(¯τ)− /a\}b∇acketle{t∇ YY,∇L/a\}b∇acket∇i}ht(¯τ) ≤δ2 Y(L)−2√ ¯τ/a\}b∇acketle{t∇YY,X/a\}b∇acket∇i}ht(¯τ) =/integraldisplay¯τ 0√τ[2|∇XY|2+ 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht+∇Y∇YR + 2∇XRic (Y,Y)−4∇YRic (Y,X)]dτ =/integraldisplay¯τ 0√τ/bracketleftbigg 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−Ric (Y,·) +1 2τY/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 + 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht+∇Y∇YR + 2∇XRic (Y,Y)−4∇YRic (Y,X)/bracketrightbigg dτ =/integraldisplay¯τ 0√τ/bracketleftbigg 2|Ric(Y,·)|2−2 τRic (Y,Y) +1 2τ¯τ+ 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht +∇Y∇YR+ 2∇XRic (Y,Y)−4∇YRic (Y,X)/bracketrightbigg dτ. Since d dτRic (Y,Y) = Ric τ(Y,Y) +∇XRic(Y,Y) + 2Ric( ∇XY,Y) = Ric τ(Y,Y) +∇XRic(Y,Y)−2|Ric(Y,·)|2+1 τRic (Y,Y), THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 249 we have Hess L(Y,Y) ≤/integraldisplay¯τ 0√τ/bracketleftbigg 2|Ric (Y,·)|2−2 τRic (Y,Y) +1 2τ¯τ+ 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht +∇Y∇YR−4(∇YRic )(X,Y)−/parenleftbigg 2d dτRic (Y,Y)−2Ric τ(Y,Y) + 4|Ric(Y,·)|2−2 τRic (Y,Y)/parenrightbigg + 4∇XRic (Y,Y)/bracketrightbigg dτ =−/integraldisplay¯τ 0/bracketleftbigg 2√τd dτRic (Y,Y) +1√τRic (Y,Y)/bracketrightbigg dτ+1 2¯τ/integraldisplay¯τ 01√τdτ +/integraldisplay¯τ 0√τ/bracketleftbigg 2/a\}b∇acketle{tR(Y,X)Y,X/a\}b∇acket∇i}ht+∇Y∇YR+1 τRic (Y,Y) + 4(∇XRic (Y,Y)− ∇ YRic (X,Y)) + 2Ric τ(Y,Y)−2|Ric(Y,·)|2/bracketrightbigg dτ =1√¯τ−2√ ¯τRic (Y,Y)−/integraldisplay¯τ 0√τQ(X,Y)dτ. This proves the inequality. As usual, the quadratic form I(V,V) =/integraldisplay¯τ 0√τ[2|∇XV|2+ 2/a\}b∇acketle{tR(V,X)V,X/a\}b∇acket∇i}ht+∇V∇VR +2∇XRic (V,V)−4∇VRic (V,X)]dτ, for any vector ﬁeld Valongγ, is called the index form. Since γis shortest, the standard Dirichlet principle for I(V,V) implies that the equality holds if and only if the vector ﬁeld Yis the derivative of a variation of γbyL-geodesics. Corollary 3.2.4. We have ∆L≤n√¯τ−2√ ¯τR−1 ¯τK in the sense of distribution. Moreover, the equality holds i f and only if we are on a gradient shrinking soliton with Rij+1 2√¯τ∇i∇jL=1 2¯τgij. Proof. Choose an orthonormal basis {Y1,...,Y n}atτ= ¯τand extend them along the shortest L-geodesicγto get vector ﬁelds Yi(τ),i= 1,...,n , by solving the ODE (3.2.8), with /a\}b∇acketle{tYi(τ),Yj(τ)/a\}b∇acket∇i}ht=τ ¯τδijon [0,¯τ]. TakingY=Yiin Proposition 3.2.3 and 250 H.-D. CAO AND X.-P. ZHU summing over i, we get ∆L≤n√¯τ−2√ ¯τR−n/summationdisplay i=1/integraldisplay¯τ 0√τQ(X,Y i)dτ (3.2.10) =n√¯τ−2√ ¯τR−/integraldisplay¯τ 0√τ/parenleftigτ ¯τ/parenrightig Q(X)dτ =n√¯τ−2√ ¯τR−1 ¯τK. Moreover, by Proposition 3.2.3, the equality in (3.2.10) ho lds everywhere if and only if for each ( q,¯τ) and any shortest L-geodesicγon [0,¯τ] connecting pandq, and for any unit vector Yatτ= ¯τ, the extended vector ﬁeld Y(τ) alongγby the ODE (3.2.8) must be an L-Jacobian ﬁeld. When Yi(τ), i= 1,...,n areL-Jacobian ﬁelds along γ, we have d dτ/a\}b∇acketle{tYi(τ),Yj(τ)/a\}b∇acket∇i}ht = 2Ric(Yi,Yj) +/a\}b∇acketle{t∇XYi,Yj/a\}b∇acket∇i}ht+/a\}b∇acketle{tYi,∇XYj/a\}b∇acket∇i}ht = 2Ric(Yi,Yj) +/angbracketleftbigg ∇Yi/parenleftbigg1 2√¯τ∇L/parenrightbigg ,Yj/angbracketrightbigg +/angbracketleftbigg Yi,∇Yj/parenleftbigg1 2√¯τ∇L/parenrightbigg/angbracketrightbigg = 2Ric(Yi,Yj) +1√¯τHess L(Yi,Yj) and then by (3.2.9), 2Ric(Yi,Yj) +1√¯τHess L(Yi,Yj) =1 ¯τδij,atτ= ¯τ. Therefore the equality in (3.2.10) holds everywhere if and o nly if we are on a gradient shrinking soliton with Rij+1 2√¯τ∇i∇jL=1 2¯τgij. In summary, from (3.2.6), (3.2.7) and Corollary 3.2.4, we ha ve   ∂L ∂¯τ= 2√¯τR−L 2¯τ+K ¯τ, |∇L|2=−4¯τR+2√¯τL−4√¯τK, ∆L≤ −2√¯τR+n√¯τ−K ¯τ, in the sense of distributions. Now the Li-Yau-Perelman distance l=l(q,¯τ) is deﬁned by l(q,¯τ) =L(q,¯τ)/2√ ¯τ. We thus have the following THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 251 Lemma 3.2.5. For the Li-Yau-Perelman distance l(q,¯τ)deﬁned above, we have ∂l ∂¯τ=−l ¯τ+R+1 2¯τ3/2K, (3.2.11) |∇l|2=−R+l ¯τ−1 ¯τ3/2K, (3.2.12) ∆l≤ −R+n 2¯τ−1 2¯τ3/2K, (3.2.13) in the sense of distributions. Moreover, the equality in (3. 2.13) holds if and only if we are on a gradient shrinking soliton. As the ﬁrst consequence, we derive the following upper bound on the minimum ofl(·,τ) for everyτwhich will be useful in proving the no local collapsing theor em in the next section. Corollary 3.2.6. Letgij(τ),τ≥0, be a family of metrics evolving by the Ricci ﬂow∂ ∂τgij= 2Rijon a compact n-dimensional manifold M. Fix a point pinMand letl(q,τ)be the Li-Yau-Perelman distance from (p,0). Then for all τ, min{l(q,τ)|q∈M} ≤n 2. Proof. Let ¯L(q,τ) = 4τl(q,τ). Then, it follows from (3.2.11) and (3.2.13) that ∂¯L ∂τ= 4τR+2K√τ, and ∆¯L≤ −4τR+ 2n−2K√τ. Hence ∂¯L ∂τ+ ∆¯L≤2n. Thus, by a standard maximum principle argument, min {¯L(q,τ)−2nτ|q∈M}is nonincreasing and therefore min {¯L(q,τ)|q∈M} ≤2nτ. As another consequence of Lemma 3.2.5, we obtain ∂l ∂¯τ−∆l+|∇l|2−R+n 2¯τ≥0. or equivalently /parenleftbigg∂ ∂¯τ−∆ +R/parenrightbigg/parenleftbig (4π¯τ)−n 2exp(−l)/parenrightbig ≤0. 252 H.-D. CAO AND X.-P. ZHU IfMis compact, we deﬁne Perelman’s reduced volume by ˜V(τ) =/integraldisplay M(4πτ)−n 2exp(−l(q,τ))dVτ(q), wheredVτdenotes the volume element with respect to the metric gij(τ). Note that Perelman’s reduced volume resembles the expression in Huis ken’s monotonicity for- mula for the mean curvature ﬂow [72]. It follows, from the abo ve computation, that d d¯τ/integraldisplay M(4π¯τ)−n 2exp(−l(q,¯τ))dV¯τ(q) =/integraldisplay M/bracketleftbigg∂ ∂¯τ((4π¯τ)−n 2exp(−l(q,¯τ))) +R(4π¯τ)−n 2exp(−l(q,¯τ))/bracketrightbigg dV¯τ(q) ≤/integraldisplay M∆((4π¯τ)−n 2exp(−l(q,¯τ)))dV¯τ(q) = 0. This says that if Mis compact, then Perelman’s reduced volume ˜V(τ) is nonincreasing inτ; moreover, the monotonicity is strict unless we are on a grad ient shrinking soliton. In order to deﬁne and to obtain the monotonicity of Perelman’ s reduced volume for a complete noncompact manifold, we need to formulate the monotonicity of Perel- man’s reduced volume in a local version. This local version i s very important and will play a crucial role in the analysis of the Ricci ﬂow with surge ry in Chapter 7. We deﬁne the L-exponential map (with parameter ¯τ)Lexp(¯τ) :TpM→M as follows: for any X∈TpM, we set LexpX(¯τ) =γ(¯τ) whereγ(τ) is the L-geodesic, starting at pand having Xas the limit of√τ˙γ(τ) as τ→0+. The associated Jacobian of the L-exponential map is called L-Jacobian . We denote by J(τ) theL-Jacobian of Lexp(τ) :TpM→M. We can now deduce an estimate for the L-Jacobian as follows. Letq=LexpX(¯τ) andγ(τ),τ∈[0,¯τ], be the shortest L-geodesic connecting p andqwith√τ˙γ(τ)→Xasτ→0+. For any vector v∈TpM, we consider the family ofL-geodesics: γs(τ) =Lexp(X+sv)(τ),0≤τ≤¯τ, s∈(−ǫ,ǫ). The associated variation vector ﬁeld V(τ), 0≤τ≤¯τ, is an L-Jacobian ﬁeld with V(0) = 0 and V(τ) = (LexpX(τ))∗(v). Letv1,...,v nbenlinearly independent vectors in TpM. Then Vi(τ) = (LexpX(τ))∗(vi), i= 1,2,...,n, arenL-Jacobian ﬁelds along γ(τ),τ∈[0,¯τ]. The L-Jacobian J(τ) is given by J(τ) =|V1(τ)∧ ··· ∧Vn(τ)|gij(τ)/|v1∧ ··· ∧vn|. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 253 Now for ﬁxed b∈(0,¯τ), we can choose linearly independent vectors v1,...,v n∈ TpMsuch that /a\}b∇acketle{tVi(b),Vj(b)/a\}b∇acket∇i}htgij(b)=δij. We compute d dτJ2 =2 |v1∧ ··· ∧vn|2n/summationdisplay j=1/a\}b∇acketle{tV1∧ ··· ∧ ∇ XVj∧ ··· ∧Vn,V1∧ ··· ∧Vn/a\}b∇acket∇i}htgij(τ) +2 |v1∧ ··· ∧vn|2n/summationdisplay j=1/a\}b∇acketle{tV1∧ ··· ∧ Ric (Vj,·)∧ ··· ∧Vn,V1∧ ··· ∧Vn/a\}b∇acket∇i}htgij(τ). Atτ=b, d dτJ2(b) =2 |v1∧ ··· ∧vn|2n/summationdisplay j=1(/a\}b∇acketle{t∇XVj,Vj/a\}b∇acket∇i}htgij(b)+ Ric (Vj,Vj)). Thus, d dτlogJ(b) =n/summationdisplay j=1(/a\}b∇acketle{t∇XVj,Vj/a\}b∇acket∇i}htgij(b)+ Ric (Vj,Vj)) =n/summationdisplay j=1/parenleftigg/angbracketleftbigg ∇Vj/parenleftbigg1 2√ b∇L/parenrightbigg ,Vj/angbracketrightbigg gij(b)+ Ric (Vj,Vj)/parenrightigg =1 2√ b n/summationdisplay j=1Hess L(Vj,Vj) +R =1 2√ b∆L+R. Therefore, in view of Corollary 3.2.4, we obtain the followi ng estimate for L-Jacobian: (3.2.14)d dτlogJ(τ)≤n 2τ−1 2τ3/2Kon [0,¯τ]. On the other hand, by the deﬁnition of the Li-Yau-Perelman di stance and (3.2.4), we have d dτl(τ) =−1 2τl+1 2√τd dτL (3.2.15) =−1 2τl+1 2√τ(√τ(R+|X|2)) =−1 2τ3/2K. Here and in the following we denote by l(τ) =l(γ(τ),τ). Now the combination of (3.2.14) and (3.2.15) implies the following important Jacobian comparison theo- remof Perelman [103]. Theorem 3.2.7 ( Perelman’s Jacobian comparison theorem ).Letgij(τ)be a family of complete solutions to the Ricci ﬂow∂ ∂τgij= 2Rijon a manifold Mwith 254 H.-D. CAO AND X.-P. ZHU bounded curvature. Let γ: [0,¯τ]→Mbe a shortest L-geodesic starting from a ﬁxed pointp. Then Perelman’s reduced volume element (4πτ)−n 2exp(−l(τ))J(τ) is nonincreasing in τalongγ. We now show how to integrate Perelman’s reduced volume eleme nt overTpMto deduce the following monotonicity result of Perelman [103] . Theorem 3.2.8 ( Monotonicity of Perelman’s reduced volume ).Letgijbe a family of complete metrics evolving by the Ricci ﬂow∂ ∂τgij= 2Rijon a manifold M with bounded curvature. Fix a point pinMand letl(q,τ)be the reduced distance from(p,0). Then (i) Perelman’s reduced volume ˜V(τ) =/integraldisplay M(4πτ)−n 2exp(−l(q,τ))dVτ(q) is ﬁnite and nonincreasing in τ; (ii) the monotonicity is strict unless we are on a gradient sh rinking soliton. Proof. For anyv∈TpMwe can ﬁnd an L-geodesicγ(τ), starting at p, with lim τ→0+√τ˙γ(τ) =v. Recall that γ(τ) satisﬁes the L-geodesic equation ∇˙γ(τ)˙γ(τ)−1 2∇R+1 2τ˙γ(τ) + 2Ric(˙γ(τ),·) = 0. Multiplying this equation by√τ, we get (3.2.16)d dτ(√τ˙γ)−1 2√τ∇R+ 2Ric(√τ˙γ(τ),·) = 0. Since the curvature of the metric gij(τ) is bounded, it follows from Shi’s derivative estimate (Theorem 1.4.1) that |∇R|is also bounded for small τ >0. Thus by inte- grating (3.2.16), we have (3.2.17) |√τ˙γ(τ)−v| ≤Cτ(|v|+ 1) forτsmall enough and for some positive constant Cdepending only the curvature bound. Letv1,...,v nbenlinearly independent vectors in TpMand let Vi(τ) = (Lexpv(τ))∗(vi) =d ds|s=0Lexp(v+svi)(τ), i= 1,...,n. TheL-Jacobian J(τ) is given by J(τ) =|V1(τ)∧ ··· ∧Vn(τ)|gij(τ)/|v1∧ ··· ∧vn| By (3.2.17), we see that /vextendsingle/vextendsingle/vextendsingle/vextendsingle√τd dτLexp(v+svi)(τ)−(v+svi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cτ(|v|+|vi|+ 1) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 255 forτsmall enough and all s∈(−ǫ,ǫ) (for some ǫ >0 small) and i= 1,...,n . This implies that lim τ→0+√τ˙Vi(τ) =vi, i= 1,...,n, so we deduce that (3.2.18) lim τ→0+τ−n 2J(τ) = 1. Meanwhile, by using (3.2.17), we have l(τ) =1 2√τ/integraldisplayτ 0√τ(R+|˙γ(τ)|2)dτ → |v|2asτ→0+. Thus (3.2.19) l(0) =|v|2. Combining (3.2.18) and (3.2.19) with Theorem 3.2.7, we get ˜V(τ) =/integraldisplay M(4πτ)−n 2exp(−l(q,τ))dVτ(q) ≤/integraldisplay TpM(4πτ)−n 2exp(−l(τ))J(τ)|τ=0dv = (4π)−n 2/integraldisplay Rnexp(−|v|2)dv <+∞. This proves that Perelman’s reduced volume is always ﬁnite a nd hence well deﬁned. Now the monotonicity assertion in (i) follows directly from Theorem 3.2.7. For the assertion (ii), we note that the equality in (3.2.13) holds everywhere when the monotonicity of Perelman’s reduced volume is not st rict. Therefore we have completed the proof of the theorem. 3.3. No Local Collapsing Theorem I. In this section we apply the monotonicity of Perelman’s reduced volume in Theorem 3.2.8 to prove Perelman’s no local collapsing theorem I , which is extremely important not only because it gives a local injectivity radius estimate in terms of local c urvature bound but also it will survive the surgeries in Chapter 7. Definition 3.3.1. Letκ,rbe two positive constants and let gij(t),0≤t< T, be a solution to the Ricci ﬂow on an n-dimensional manifold M. We call the solution gij(t)κ-noncollapsed at (x0,t0)∈M×[0,T) on the scale rif it satisﬁes the following property: whenever |Rm|(x,t)≤r−2 for allx∈Bt0(x0,r) andt∈[t0−r2,t0], we have Volt0(Bt0(x0,r))≥κrn. 256 H.-D. CAO AND X.-P. ZHU HereBt0(x0,r) is the geodesic ball centered at x0∈Mand of radius rwith respect to the metric gij(t0). Now we are ready to state the no local collapsing theorem I of Perelman [103]. Theorem 3.3.2 ( No local collapsing theorem I ).Given any metric gijon an n-dimensional compact manifold M. Letgij(t)be the solution to the Ricci ﬂow on [0,T), withT <+∞, starting at gij. Then there exist positive constants κandρ0such that for any t0∈[0,T)and any point x0∈M, the solution gij(t)isκ-noncollapsed at(x0,t0)on all scales less than ρ0. Proof. We argue by contradiction. Suppose that there are sequences pk∈M, tk∈[0,T) andrk→0 such that (3.3.1) |Rm|(x,t)≤r−2 k forx∈Bk=Btk(pk,rk) andtk−r2 k≤t≤tk, but (3.3.2) ǫk=r−1 kVoltk(Bk)1 n→0 ask→ ∞. Without loss of generality, we may assume that tk→Task→+∞. Let ¯τ(t) =tk−t,p=pkand ˜Vk(¯τ) =/integraldisplay M(4π¯τ)−n 2exp(−l(q,¯τ))dVtk−¯τ(q), wherel(q,¯τ) is the Li-Yau-Perelman distance with respect to p=pk. Step1. We ﬁrst want to show that for klarge enough, ˜Vk(ǫkr2 k)≤2ǫn 2 k. For anyv∈TpMwe can ﬁnd an L-geodesicγ(τ) starting at pwith lim τ→0√τ˙γ(τ) =v. Recall that γ(τ) satisﬁes the equation (3.2.16). It follows from assumptio n (3.3.1) and Shi’s local derivative estimate (Theorem 1.4.2 ) that |∇R|has a bound in the order of 1 /r3 kfort∈[tk−ǫkr2 k,tk]. Thus by integrating (3.2.16) we see that for τ≤ǫkr2 ksatisfying the property that γ(σ)∈Bkas long asσ<τ, there holds (3.3.3) |√τ˙γ(τ)−v| ≤Cǫk(|v|+ 1) whereCis some positive constant depending only on the dimension. H ere we have implicitly used the fact that the metric gij(t) is equivalent for x∈Bkandt∈[tk− ǫkr2 k,tk]. In fact since∂gij ∂t=−2Rijand|Rm| ≤r−2 konBk×[tk−r2 k,tk], we have (3.3.4) e−2ǫkgij(x,tk)≤gij(x,t)≤e2ǫkgij(x,tk), forx∈Bkandt∈[tk−ǫkr2 k,tk]. Supposev∈TpMwith|v| ≤1 4ǫ−1 2 k. Letτ≤ǫkr2 ksuch thatγ(σ)∈Bkas long as σ<τ, whereγis theL-geodesic starting at pwith lim τ→0√τ˙γ(τ) =v. Then, by (3.3.3) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 257 and (3.3.4), for klarge enough, dtk(pk,γ(τ))≤/integraldisplayτ 0|˙γ(σ)|gij(tk)dσ <1 2ǫ−1 2 k/integraldisplayτ 0dσ√σ =ǫ−1 2 k√τ ≤rk. This shows that for klarge enough, (3.3.5) Lexp{|v|≤1 4ǫ−1/2 k}(ǫkr2 k)⊂Bk=Btk(pk,rk). We now estimate the integral of ˜Vk(ǫkr2 k) as follows, ˜Vk(ǫkr2 k) =/integraldisplay M(4πǫkr2 k)−n 2exp(−l(q,ǫkr2 k))dVtk−ǫkr2 k(q)(3.3.6) =/integraldisplay Lexp {|v|≤1 4ǫ−1/2 k}(ǫkr2 k)(4πǫkr2 k)−n 2exp(−l(q,ǫkr2 k))dVtk−ǫkr2 k(q) +/integraldisplay M\Lexp {|v|≤1 4ǫ−1/2 k}(ǫkr2 k)(4πǫkr2 k)−n 2exp(−l(q,ǫkr2 k))dVtk−ǫkr2 k(q). We observe that for each q∈Bk, L(q,ǫkr2 k) =/integraldisplayǫkr2 k 0√τ(R+|˙γ|2)dτ≥ −C(n)r−2 k(ǫkr2 k)3 2=−C(n)ǫ3 2 krk, hencel(q,ǫkr2 k)≥ −C(n)ǫk.Thus, the ﬁrst term on the RHS of (3.3.6) can be esti- mated by /integraldisplay Lexp {|v|≤1 4ǫ−1/2 k}(ǫkr2 k)(4πǫkr2 k)−n 2exp(−l(q,ǫkr2 k))dVtk−ǫkr2 k(q) (3.3.7) ≤enǫk/integraldisplay Bk(4πǫkr2 k)−n 2exp(−l(q,ǫkr2 k))dVtk(q) ≤enǫk(4π)−n 2·eC(n)ǫk·ǫ−n 2 k·(r−n kVoltk(Bk)) =e(n+C(n))ǫk(4π)−n 2·ǫn 2 k, where we have also used (3.3.5) and (3.3.4). Meanwhile, by using (3.2.18), (3.2.19) and the Jacobian Com parison Theorem 258 H.-D. CAO AND X.-P. ZHU 3.2.7, the second term on the RHS of (3.3.6) can be estimated a s follows /integraldisplay M\Lexp {|v|≤1 4ǫ−1 2 k}(ǫkr2 k)(4πǫkr2 k)−n 2exp(−l(q,ǫkr2 k))dVtk−ǫkr2 k(q) (3.3.8) ≤/integraldisplay {|v|>1 4ǫ−1 2 k}(4πτ)−n 2exp(−l(τ))J(τ)|τ=0dv = (4π)−n 2/integraldisplay {|v|>1 4ǫ−1 2 k}exp(−|v|2)dv ≤ǫn 2 k, forksuﬃciently large. Combining (3.3.6)-(3.3.8), we ﬁnish the proof of Step 1. Step2. We next want to show ˜Vk(tk) = (4πtk)−n 2/integraldisplay Mexp(−l(q,tk))dV0(q)>C′ for allk, whereC′is some positive constant independent of k. It suﬃces to show the Li-Yau-Perelman distance l(·,tk) is uniformly bounded from above onM. By Corollary 3.2.6 we know that the minimum of l(·,τ) does not exceed n 2for eachτ >0. Chooseqk∈Msuch that the minimum of l(·,tk−T 2) is attained at qk. We now construct a path γ: [0,tk]→Mconnecting pkto any given point q∈M as follows: the ﬁrst half path γ|[0,tk−T 2]connectspktoqkso that l/parenleftbigg qk,tk−T 2/parenrightbigg =1 2/radicalig tk−T 2/integraldisplaytk−T 2 0√τ(R+|˙γ(τ)|2)dτ≤n 2 and the second half path γ|[tk−T 2,tk]is a shortest geodesic connecting qktoqwith respect to the initial metric gij(0). Then, for any q∈Mn, l(q,tk) =1 2√tkL(q,tk) ≤1 2√tk/parenleftigg/integraldisplaytk−T 2 0+/integraldisplaytk tk−T 2/parenrightigg √τ(R+|˙γ(τ)|2)dτ ≤1 2√tk/parenleftigg n/radicalbigg tk−T 2+/integraldisplaytk tk−T 2√τ(R+|˙γ(τ)|2)dτ/parenrightigg ≤C for some constant C >0, since all geometric quantities in gijare uniformly bounded whent∈[0,T 2] (or equivalently, τ∈[tk−T 2,tk]). Combining Step 1 with Step 2, and using the monotonicity of ˜Vk(τ), we get C′<˜Vk(tk)≤˜Vk(ǫkr2 k)≤2ǫn 2 k→0 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 259 ask→ ∞ . This gives the desired contradiction. Therefore we have pr oved the theorem. The above no local collapsing theorem I says that if |Rm| ≤r−2on the parabolic ball{(x,t)|dt0(x,x0)≤r, t0−r2≤t≤t0}, then the volume of the geodesic ballBt0(x0,r) (with respect to the metric gij(t0)) is bounded from below by κrn. In [103], Perelman used the monotonicity of the W-functional (deﬁned by (1.5.9)) to obtain a stronger version of the no local collapsing theor em, where the curvature bound assumption on the parabolic ball is replaced by that on the geodesic ball Bt0(x0,r). The following result, called no local collapsing theorem I′, gives a further extension where the bound on the curvature tensor is replaced by the bound on the scalar curvature only. We now follow a clever argument by Bing-Long Chen. Theorem 3.3.3 ( No local collapsing theorem I′).SupposeMis a compact Rie- mannian manifold, and gij(t),0≤t<T < +∞, is a solution to the Ricci ﬂow. Then there exists a positive constant κdepending only the initial metric and Tsuch that for any (x0,t0)∈M×(0,T)if R(x,t0)≤r−2,∀x∈Bt0(x0,r) with0<r≤√ T, then we have Volt0(Bt0(x0,r))≥κrn. Proof. We will prove the assertion (∗)a Volt0(Bt0(x0,a))≥κan for all 0<a≤r. Recall that µ(gij,τ) = inf/braceleftbigg W(gij,f,τ)/vextendsingle/vextendsingle/vextendsingle/integraldisplay M(4πτ)−n 2e−fdV= 1/bracerightbigg . Set µ0= inf 0≤τ≤2Tµ(gij(0),τ)>−∞. By Corollary 1.5.9, we have µ(gij(t0),b)≥µ(gij(0),t0+b) (3.3.9) ≥µ0 for 0<b≤r2. Let 0<ζ≤1 be a positive smooth function on Rwhereζ(s) = 1 for |s| ≤1 2,|ζ′|2/ζ≤20 everywhere, and ζ(s) is very close to zero for |s| ≥1. Deﬁne a functionfonMby (4πr2)−n 2e−f(x)=e−c(4πr2)−n 2ζ/parenleftbiggdt0(x,x0) r/parenrightbigg , where the constant cis chosen so that/integraltext M(4πr2)−n 2e−fdVt0= 1. Then it follows from (3.3.9) that W(gij(t0),f,r2) =/integraldisplay M[r2(|∇f|2+R) +f−n](4πr2)−n 2e−fdVt0 (3.3.10) ≥µ0. 260 H.-D. CAO AND X.-P. ZHU Note that 1 =/integraldisplay M(4πr2)−n 2e−cζ/parenleftbiggdt0(x,x0) r/parenrightbigg dVt0 ≥/integraldisplay Bt0(x0,r 2)(4πr2)−n 2e−cdVt0 = (4πr2)−n 2e−cVolt0/parenleftig Bt0/parenleftig x0,r 2/parenrightig/parenrightig . By combining with (3.3.10) and the scalar curvature bound, w e have c≥ −/integraldisplay M/parenleftbigg(ζ′)2 ζ−logζ·ζ/parenrightbigg e−c(4πr2)−n 2dVt0+ (n−1) +µ0 ≥ −2(20 +e−1)e−c(4πr2)−n 2Volt0(Bt0(x0,r)) + (n−1) +µ0 ≥ −2(20 +e−1)Volt0(Bt0(x0,r)) Volt0(Bt0(x0,r 2))+ (n−1) +µ0, where we used the fact that ζ(s) is very close to zero for |s| ≥1. Note also that 2/integraldisplay Bt0(x0,r)e−c(4πr2)−n 2dVt0≥/integraldisplay M(4πr2)−n 2e−fdVt0= 1. Let us set κ= min/braceleftbigg1 2exp(−2(20 +e−1)3−n+ (n−1) +µ0),1 2αn/bracerightbigg whereαnis the volume of the unit ball in Rn. Then we obtain Volt0(Bt0(x0,r))≥1 2ec(4πr2)n 2 ≥1 2(4π)n 2exp(−2(20 +e−1)3−n+ (n−1) +µ0)·rn ≥κrn provided Vol t0(Bt0(x0,r 2))≥3−nVolt0(Bt0(x0,r)). Note that the above argument also works for any smaller radiu sa≤r. Thus we have proved the following assertion: (3.3.11) Vol t0(Bt0(x0,a))≥κan whenevera∈(0,r] and Vol t0(Bt0(x0,a 2))≥3−nVolt0(Bt0(x0,a)). Now we argue by contradiction to prove the assertion ( ∗)afor anya∈(0,r]. Suppose ( ∗)afails for some a∈(0,r]. Then by (3.3.11) we have Volt0(Bt0(x0,a 2))<3−nVolt0(Bt0(x0,a)) <3−nκan <κ/parenleftiga 2/parenrightign . This says that ( ∗)a 2would also fail. By induction, we deduce that Volt0/parenleftig Bt0/parenleftig x0,a 2k/parenrightig/parenrightig <κ/parenleftiga 2k/parenrightign for allk≥1. This is a contradiction since lim k→∞Volt0/parenleftbig Bt0/parenleftbig x0,a 2k/parenrightbig/parenrightbig //parenleftbiga 2k/parenrightbign=αn. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 261 3.4. No Local Collapsing Theorem II. By inspecting the arguments in the previous section, one can see that if the injectivity radius of the initial metric is uniformly bounded from below, then the no local collapsing t heorem I also holds for complete solutions with bounded curvature on a complete non compact manifold. In this section we will use a cut-oﬀ argument to extend the no loc al collapsing theorem to any complete solution with bounded curvature. In some sen se, the second no local collapsing theorem gives a good relative estimate of the vol ume element for the Ricci ﬂow. We ﬁrst need the following useful lemma which contains two as sertions. The ﬁrst one is a parabolic version of the Laplacian comparison theor em (where the curvature sign restriction in the ordinary Laplacian comparison is es sentially removed in the Ricci ﬂow). The second one is a generalization of a result of H amilton (Theorem 17.2 in [63]), where it was derived by an integral version of Bonne t-Myers’ theorem. Lemma 3.4.1 ( Perelman [103] ).Letgij(x,t)be a solution to the Ricci ﬂow on ann-dimensional manifold Mand denote by dt(x,x0)the distance between xandx0 with respect to the metric gij(t). (i) Suppose Ric (·,t0)≤(n−1)KonBt0(x0,r0)for somex0∈Mand some positive constants K and r0. Then the distance function d(x,t) =dt(x,x0) satisﬁes, at t=t0and outside Bt0(x0,r0), the diﬀerential inequality: ∂ ∂td−∆d≥ −(n−1)/parenleftbigg2 3Kr0+r−1 0/parenrightbigg . (ii) Suppose Ric (·,t0)≤(n−1)KonBt0(x0,r0)/uniontextBt0(x1,r0)for somex0,x1∈ Mand some positive constants K and r0. Then, at t=t0, d dtdt(x0,x1)≥ −2(n−1)/parenleftbigg2 3Kr0+r−1 0/parenrightbigg . Proof. Letγ: [0,d(x,t0)]→Mbe a shortest normal geodesic from x0tox with respect to the metric gij(t0). As usual, we may assume that xandx0are not conjugate to each other in the metric gij(t0), otherwise we can understand the diﬀerential inequality in the barrier sense. Let X= ˙γ(0) and let {X,e1,...,e n−1}be an orthonormal basis of Tx0M. Extend this basis parallel along γto form a parallel orthonormal basis {X(s),e1(s),...,e n−1(s)}alongγ. (i) LetXi(s), i= 1,...,n −1, be the Jacobian ﬁelds along γsuch thatXi(0) = 0 andXi(d(x,t0)) =ei(d(x,t0)) fori= 1,...,n −1. Then it is well-known that (see for example [112]) ∆dt0(x,x0) =n−1/summationdisplay i=1/integraldisplayd(x,t0) 0(|˙Xi|2−R(X,X i,X,X i))ds (in Proposition 3.2.3 we actually did this for the more compl icated L-distance func- tion). Deﬁne vector ﬁelds Yi, i= 1,...,n −1, alongγas follows: Yi(s) =/braceleftigg s r0ei(s),ifs∈[0,r0], ei(s),ifs∈[r0,d(x,t0)]. 262 H.-D. CAO AND X.-P. ZHU which have the same value as the corresponding Jacobian ﬁeld sXi(s) at the two end points ofγ. Then by using the standard index comparison theorem (see fo r example [22]) we have ∆dt0(x,x0) =n−1/summationdisplay i=1/integraldisplayd(x,t0) 0(|˙Xi|2−R(X,X i,X,X i))ds ≤n−1/summationdisplay i=1/integraldisplayd(x,t0) 0(|˙Yi|2−R(X,Y i,X,Y i))ds =/integraldisplayr0 01 r2 0(n−1−s2Ric (X,X))ds+/integraldisplayd(x,t0) r0(−Ric (X,X))ds =−/integraldisplay γRic (X,X) +/integraldisplayr0 0/parenleftbigg(n−1) r2 0+/parenleftbigg 1−s2 r2 0/parenrightbigg Ric (X,X)/parenrightbigg ds ≤ −/integraldisplay γRic (X,X) + (n−1)/parenleftbigg2 3Kr0+r−1 0/parenrightbigg . On the other hand, ∂ ∂tdt(x,x0) =∂ ∂t/integraldisplayd(x,t0) 0/radicalig gijXiXjds =−/integraldisplay γRic(X,X)ds. Hence we obtain the desired diﬀerential inequality. (ii) The proof is divided into three cases. Case(1):dt0(x0,x1)≥2r0. Deﬁne vector ﬁelds Yi, i= 1,...,n −1, alongγas follows: Yi(s) =  s r0ei(s), ifs∈[0,r0], ei(s), ifs∈[r0,d(x1,t0)], d(x1,t0)−s r0ei(s),ifs∈[d(x1,t0)−r0,d(x1,t0)]. Then by the second variation formula, we have n−1/summationdisplay i=1/integraldisplayd(x1,t0) 0R(X,Y i,X,Y i)ds≤n−1/summationdisplay i=1/integraldisplayd(x1,t0) 0|˙Yi|2ds, which implies /integraldisplayr0 0s2 r2 0Ric (X,X)ds+/integraldisplayd(x,t0)−r0 r0Ric(X,X)ds +/integraldisplayd(x1,t0) d(x1,t0)−r0/parenleftbiggd(x1,t0)−s r0/parenrightbigg2 Ric (X,X)ds≤2(n−1) r0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 263 Thus d dt(dt(x0,x1)) ≥ −/integraldisplayr0 0/parenleftbigg 1−s2 r2 0/parenrightbigg Ric (X,X)ds −/integraldisplayd(x1,t0) d(x1,t0)−r0/parenleftigg 1−/parenleftbiggd(x1,t0)−s r0/parenrightbigg2/parenrightigg Ric (X,X)ds−2(n−1) r0 ≥ −2(n−1)/parenleftbigg2 3Kr0+r−1 0/parenrightbigg . Case(2):2√ 2K 3≤dt0(x0,x1)≤2r0. In this case, letting r1=1√ 2K 3and applying case (1) with r0replaced by r1,we get d dt(dt(x0,x1))≥ −2(n−1)/parenleftbigg2 3Kr1+r−1 1/parenrightbigg ≥ −2(n−1)/parenleftbigg2 3Kr0+r−1 0/parenrightbigg . Case(3):dt0(x0,x1)≤min/braceleftig 2√ 2K 3,2r0/bracerightig . In this case, /integraldisplayd(x1,t0) 0Ric (X,X)ds≤(n−1)K2/radicalig 2K 3= (n−1)√ 6K, and 2(n−1)/parenleftbigg2 3Kr0+r−1 0/parenrightbigg ≥(n−1)/radicalbigg 32 3K. This proves the lemma. The following result, called the no local collapsing theorem II , was obtained by Perelman in [103]. Theorem 3.4.2 ( No local collapsing theorem II ).For anyA >0there exists κ=κ(A)>0with the following property: if gij(t)is a complete solution to the Ricci ﬂow on 0≤t≤r2 0with bounded curvature and satifying |Rm|(x,t)≤r−2 0on B 0(x0,r0)×[0,r2 0] and Vol0(B0(x0,r0))≥A−1rn 0, thengij(t)isκ-noncollapsed on all scales less than r0at every point (x,r2 0)with dr2 0(x,x0)≤Ar0. 264 H.-D. CAO AND X.-P. ZHU Proof. From the evolution equation of the Ricci ﬂow, we know that the metrics gij(·,t) are equivalent to each other on B0(x0,r0)×[0,r2 0]. Thus, without loss of generality, we may assume that the curvature of the solution is uniformly bounded for allt∈[0,r2 0] and all points in Bt(x0,r0). Fix a point ( x,r2 0)∈M× {r2 0}. By scaling we may assume r0= 1. We may also assume d1(x,x0) =A. Letp=x, ¯τ= 1−t, and consider Perelman’s reduced volume ˜V(¯τ) =/integraldisplay M(4π¯τ)−n 2exp(−l(q,¯τ))dV1−¯τ(q), where l(q,¯τ) = inf/braceleftbigg1 2√¯τ/integraldisplay¯τ 0√τ(R+|˙γ|2)dτ|γ: [0,¯τ]→M withγ(0) =p, γ(¯τ) =q/bracerightbigg is the Li-Yau-Perelman distance. We argue by contradiction . Suppose for some 0 < r<1 we have |Rm|(y,t)≤r−2 whenevery∈B1(x,r) and 1 −r2≤t≤1, butǫ=r−1Vol1(B1(x,r))1 nis very small. Then arguing as in the proof of the no local collapsing theore m I (Theorem 3.3.2), we see that Perelman’s reduced volume ˜V(ǫr2)≤2ǫn 2. On the other hand, from the monotonicity of Perelman’s reduc ed volume we have (4π)−n 2/integraldisplay Mexp(−l(q,1))dV0(q) =˜V(1)≤˜V(ǫr2). Thus once we bound the function l(q,1) overB0(x0,1) from above, we will get the desired contradiction and will prove the theorem. For anyq∈B0(x0,1), exactly as in the proof of the no local collapsing theorem I, we choose a path γ: [0,1]→Mwithγ(0) =x, γ(1) =q,γ(1 2) =y∈B1 2(x0,1 10) andγ(τ)∈B1−τ(x0,1) forτ∈[1 2,1] such that L(γ|[0,1 2]) = 2/radicalbigg 1 2l/parenleftbigg y,1 2/parenrightbigg /parenleftbigg =L/parenleftbigg y,1 2/parenrightbigg/parenrightbigg . NowL(γ|[1 2,1]) =/integraltext1 1 2√τ(R(γ(τ),1−τ) +|˙γ(τ)|2 gij(1−τ))dτis bounded from above by a uniform constant since all geometric quantities in gijare uniformly bounded on {(y,t)|t∈[0,1/2],y∈Bt(x0,1)}(wheret∈[0,1/2] is equivalent to τ∈[1/2,1]). Thus all we need is to estimate the minimum of l(·,1 2), or equivalently ¯L(·,1 2) = 41 2l(·,1 2), in the ball B1 2(x0,1 10). Recall that ¯Lsatisﬁes the diﬀerential inequality (3.4.1)∂¯L ∂τ+ ∆¯L≤2n. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 265 We will use this in a maximum principle argument. Let us deﬁne h(y,t) =φ(d(y,t)−A(2t−1))·(¯L(y,1−t) + 2n+ 1) whered(y,t) =dt(y,x0), andφis a function of one variable, equal to 1 on ( −∞,1 20), and rapidly increasing to inﬁnity on (1 20,1 10) in such a way that: (3.4.2) 2(φ′)2 φ−φ′′≥(2A+ 100n)φ′−C(A)φ for some constant C(A)<+∞. The existence of such a function φcan be justiﬁed as follows: put v=φ′ φ, then the condition (3.4.2) for φcan be written as 3v2−v′≥(2A+ 100n)v−C(A) which can be solved for v. Since the scalar curvature Revolves by ∂R ∂t= ∆R+ 2|Rc|2≥∆R+2 nR2, we can apply the maximum principle as in Chapter 2 to deduce R(x,t)≥ −n 2tfort∈(0,1] andx∈M. Thus for ¯τ= 1−t∈[0,1 2], ¯L(·,¯τ) = 2√ ¯τ/integraldisplay¯τ 0√τ(R+|˙γ|2)dτ ≥2√ ¯τ/integraldisplay¯τ 0√τ/parenleftbigg −n 2(1−τ)/parenrightbigg dτ ≥2√ ¯τ/integraldisplay¯τ 0√τ(−n)dτ >−2n. That is (3.4.3) ¯L(·,1−t) + 2n+ 1≥1,fort∈/bracketleftbigg1 2,1/bracketrightbigg . Clearly min y∈Mh(y,1 2) is achieved by some y∈B1 2(x0,1 10) and (3.4.4) min y∈Mh(y,1)≤h(x,1) = 2n+ 1. 266 H.-D. CAO AND X.-P. ZHU We compute /parenleftbigg∂ ∂t−∆/parenrightbigg h=/parenleftbigg∂ ∂t−∆/parenrightbigg φ·(¯L(y,1−t) + 2n+ 1) +φ·/parenleftbigg∂ ∂t−∆/parenrightbigg ¯L(y,1−t)−2/a\}b∇acketle{t∇φ,∇¯L(y,1−t)/a\}b∇acket∇i}ht =/parenleftbigg φ′/bracketleftbigg/parenleftbigg∂ ∂t−∆/parenrightbigg d−2A/bracketrightbigg −φ′′|∇d|2/parenrightbigg ·(¯L+ 2n+ 1) +φ·/parenleftbigg −∂ ∂τ−∆/parenrightbigg ¯L−2/a\}b∇acketle{t∇φ,∇¯L/a\}b∇acket∇i}ht ≥/parenleftbigg φ′/bracketleftbigg/parenleftbigg∂ ∂t−∆/parenrightbigg d−2A/bracketrightbigg −φ′′/parenrightbigg ·(¯L+ 2n+ 1) −2nφ−2/a\}b∇acketle{t∇φ,∇¯L/a\}b∇acket∇i}ht by using (3.4.1). At a minimizing point of hwe have ∇φ φ=−∇¯L (¯L+ 2n+ 1). Hence −2/a\}b∇acketle{t∇φ,∇¯L/a\}b∇acket∇i}ht= 2|∇φ|2 φ(¯L+ 2n+ 1) = 2(φ′)2 φ(¯L+ 2n+ 1). Then at the minimizing point of h, we compute /parenleftbigg∂ ∂t−∆/parenrightbigg h≥/parenleftbigg φ′/bracketleftbigg/parenleftbigg∂ ∂t−∆/parenrightbigg d−2A/bracketrightbigg −φ′′/parenrightbigg ·(¯L+ 2n+ 1) −2nφ+ 2(φ′)2 φ(¯L+ 2n+ 1) ≥/parenleftbigg φ′/bracketleftbigg/parenleftbigg∂ ∂t−∆/parenrightbigg d−2A/bracketrightbigg −φ′′/parenrightbigg ·(¯L+ 2n+ 1) −2nh+ 2(φ′)2 φ(¯L+ 2n+ 1) fort∈[1 2,1] and ∆h≥0. Let us denote by hmin(t) = min y∈Mh(y,t). By applying Lemma 3.4.1(i) to the set where φ′/\e}atio\slash= 0, we further obtain d dthmin≥(¯L+ 2n+ 1)·/bracketleftbigg φ′(−100n−2A)−φ′′+ 2(φ′)2 φ/bracketrightbigg −2nhmin ≥ −(2n+C(A))hmin,fort∈[1 2,1]. This implies that hmin(t) cannot decrease too fast. By combining (3.4.3) and (3.4.4) we get the required estimate for the minimum ¯L(·,1 2) in the ball B1 2(x0,1 10). Therefore we have completed the proof of the theorem. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 267 4. Formation of Singularities. Letgij(x,t) be a solution to the Ricci ﬂow on M×[0,T) and suppose [0 ,T),T≤ ∞, is the maximal time interval. If T <+∞, then the short time existence theorem tells us the curvature of the solution becomes unbounded as t→T. We then say the solution develops a singularity ast→T. As in the minimal surface theory and harmonic map theory, one usually tries to understand the structure of a singularity of the Ricci ﬂow by rescaling the solution (or blow up) to obtain a sequence of solutions to the Ricci ﬂow with uniformly bounded curvature on compact subsets and looking at its limit. The main purpose of this chapter is to establish a convergenc e theorem for a sequence of solutions to the Ricci ﬂow with uniform bounded c urvature on compact subsets and to use the convergence theorem to give a rough cla ssiﬁcation for singular- ities of solutions to the Ricci ﬂow. Further studies on the st ructures of singularities of the Ricci ﬂow will be given in Chapter 6 and 7. 4.1. Cheeger Type Compactness. We begin with the concept of C∞ locconver- gence of tensors on a given manifold M. LetTibe a sequence of tensors on M. We say that Ticonverges to a tensor Tin theC∞ loctopology if we can ﬁnd a covering {(Us,ϕs)},ϕs:Us→Rn, ofC∞coordinate charts so that for every compact set K⊂M, the components of Ticonverge in the C∞topology to the components of Tin the intersections of Kwith these coordinate charts, considered as functions on ϕs(Us)⊂Rn. Consider a Riemannian manifold ( M,g). Amarking onMis a choice of a pointp∈Mwhich we call the origin . We will refer to such a triple ( M,g,p ) as amarked Riemannian manifold . Definition 4.1.1. Let (Mk,gk,pk) be a sequence of marked complete Rie- mannian manifolds, with metrics gkand marked points pk∈Mk. LetB(pk,sk)⊂Mk denote the geodesic ball centered at pk∈Mkand of radius sk(0< sk≤+∞). We say a sequence of marked geodesic balls ( B(pk,sk),gk,pk) withsk→s∞(≤+∞) converges in theC∞ loctopology to a marked (maybe noncomplete) manifold (B∞,g∞,p∞), which is an open geodesic ball centered at p∞∈B∞and of radius s∞with respect to the metric g∞, if we can ﬁnd a sequence of exhausting open sets UkinB∞containingp∞and a sequence of diﬀeomorphisms fkof the setsUkinB∞ to open sets VkinB(pk,sk)⊂Mkmappingp∞topksuch that the pull-back metrics ˜gk= (fk)∗gkconverge in C∞topology to g∞on every compact subset of B∞. We remark that this concept of C∞ loc-convergence of a sequence of marked mani- folds (Mk,gk,pk) is not the same as that of C∞ loc-convergence of metric tensors on a given manifold, even when we are considering the sequence of Riemannian metric gk on the same space M. This is because one can have a sequence of diﬀeomorphisms fk:M→Msuch that ( fk)∗gkconverges in C∞ loctopology while gkitself does not converge. There have been a lot of work in Riemannian geometry on the con vergence of a sequence of compact manifolds with bounded curvature, dia meter and injectivity radius (see for example Gromov [53], Peters [106], and Green e and Wu [51]). The following theorem, which is a slight generalization of Hami lton’s convergence theorem [62], modiﬁes these results in three aspects: the ﬁrst one is to allow noncompact limits and then to avoid any diameter bound; the second one is to avoi d having to assume a uniform lower bound for the injectivity radius over the whol e manifold, a hypothesis which is much harder to satisfy in applications; the last one is to avoid a uniform curvature bound over the whole manifold so that we can take a l ocal limit. Theorem 4.1.2 ( Hamilton [62] ).Let(Mk, gk, pk)be a sequence of marked 268 H.-D. CAO AND X.-P. ZHU complete Riemannian manifolds of dimension n. Consider a sequence of geodesic balls B(pk,sk)⊂Mkof radiussk(0< sk≤ ∞), withsk→s∞(≤ ∞), around the base pointpkofMkin the metric gk. Suppose (a) for every radius r < s ∞and every integer l≥0there exists a constant Bl,r, independent of k, and positive integer k(r,l)<+∞such that as k≥ k(r,l), the curvature tensors Rm(gk)of the metrics gkand theirlth-covariant derivatives satisfy the bounds |∇lRm(gk)| ≤Bl,r on the balls B(pk,r)of radiusraroundpkin the metrics gk; and (b) there exists a constant δ >0independent of ksuch that the injectivity radii inj(Mk,pk)ofMkatpkin the metric gksatisfy the bound inj(Mk,pk)≥δ. Then there exists a subsequence of the marked geodesic balls (B(pk, sk), gk, pk)which converges to a marked geodesic ball (B(p∞,s∞),g∞,p∞)inC∞ loctopology. Moreover the limit is complete ifs∞= +∞. Proof. In [62] and Theorem 16.1 of [63], Hamilton proved this conver gence the- orem for the case s∞= +∞. In the following we only need to modify Hamilton’s argument to prove the remaining case of s∞<+∞. Suppose we are given a sequence of geodesic balls ( B(pk,sk),gk,pk)⊂(Mk,gk,pk), withsk→s∞(<+∞), satisfying the assumptions of Theorem 4.1.2. We will split the proof int o three steps. Step1: Picking the subsequence. By the local injectivity radius estimate (4.2.2) in Corolla ry 4.2.3 of the next section, we can ﬁnd a positive decreasing C1functionρ(r), 0≤r<s ∞, independent ofksuch that ρ(r)<1 100(s∞−r), (4.1.1) 0≥ρ′(r)≥ −1 1000, (4.1.2) and a sequence of positive constants εk→0 so that the injectivity radius at any point x∈B(pk,sk) withrk=dk(x,pk)<s∞−εkis bounded from below by (4.1.3) inj( Mk,x)≥500ρ(rk(x)), whererk(x) =dk(x,pk) is the distance from xtopkin the metric gkofMk. We deﬁne ˜ρ(r) =ρ(r+ 20ρ(r)),˜˜ρ(r) = ˜ρ(r+ 20˜ρ(r)). By (4.1.2) we know that both ˜ ρ(r) and ˜˜ρ(r) are nonincreasing positive functions on [0,s∞). In eachB(pk,s∞) we choose inductively a sequence of points xα kforα= 0,1,2,... in the following way. First we let x0 k=pk. Oncexα kare chosen for α= 0,1,2,...,σ , we pickxσ+1 kclosest topkso thatrσ+1 k=rk(xσ+1 k) is as small as possible, subject to the requirement that the open ball B(xσ+1 k,˜˜ρσ+1 k) aroundxσ+1 kof radius ˜˜ρσ+1 kis disjoint THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 269 from the balls B(xα k,˜˜ρα k) forα= 0,1,2,...,σ , where ˜˜ρα k=˜˜ρ(rα k) andrα k=rk(xα k). In particular, the open balls B(xα k,˜˜ρα k),α= 0,1,2,..., are all disjoint. We claim the ballsB(xα k,2˜˜ρα k) coverB(pk,s∞−εk) and moreover for any r, 0<r<s ∞−εk, we can ﬁndλ(r) independent of ksuch that for klarge enough, the geodesic balls B(xα k,2˜˜ρα k) forα≤λ(r) cover the ball B(pk,r). To see this, let x∈B(pk,s∞−εk) and letr(x) be the distance from xtopkand let˜˜ρ=˜˜ρ(r(x)). Consider those αwithrα k≤r(x)<s∞−εk. Then ˜˜ρ≤˜˜ρα k. Now the given point xmust lie in one of the balls B(xα k,2˜˜ρα k). If not, we could choose the next point in the sequence of xβ kto bexinstead, for since ˜˜ρα k+˜˜ρ≤2˜˜ρα kthe ball B(x,˜˜ρ) would miss B(xα k,˜˜ρα k) withrα k≤r(x). But this is a contradiction. Moreover for anyr, 0< r < s ∞−εk, using the curvature bound and the injectivity radius bound, each ball B(xα k,˜˜ρα k) withrα k≤rhas volume at least ǫ(r)˜˜ρnwhereǫ(r)>0 is some constant depending on rbut independent of k. Now these balls are all disjoint and contained in the ball B(pk,(r+s∞)/2). On the other hand, for large enough k, we can estimate the volume of this ball from above, again usin g the curvature bound, by a positive function of rthat is independent of k. Thus there is a k′(r)>0 such that for each k≥k′(r), there holds (4.1.4) # {α|rα k≤r} ≤λ(r) for some positive constant λ(r) depending only on r, and the geodesic balls B(xα k,2˜˜ρα k) forα≤λ(r) cover the ball B(pk,r). By the way, since rα k≤rα−1 k+˜˜ρα−1 k+˜˜ρα k ≤rα−1 k+ 2˜˜ρα−1 k, and by (4.1.1) ˜˜ρα−1 k≤1 100(s∞−rα−1 k), we get by induction rα k≤49 50rα−1 k+1 50s∞ ≤/parenleftbigg49 50/parenrightbiggα r0 k+1 50/parenleftigg 1 +49 50+···+/parenleftbigg49 50/parenrightbiggα−1/parenrightigg s∞ =/parenleftbigg 1−/parenleftbigg49 50/parenrightbiggα/parenrightbigg s∞. So for each α, withα≤λ(r) (r<s ∞), there holds (4.1.5) rα k≤/parenleftigg 1−/parenleftbigg49 50/parenrightbiggλ(r)/parenrightigg s∞ for allk. And by passing to a subsequence (using a diagonalization ar gument) we may assume that rα kconverges to some rαfor eachα. Then ˜˜ρα k(respectively ˜ ρα k,ρα k) converges to ˜˜ρα=˜˜ρ(rα) (respectively ˜ ρα= ˜ρ(rα),ρα=ρ(rα)). 270 H.-D. CAO AND X.-P. ZHU Hence for all αwe can ﬁnd k(α) such that 1 2˜˜ρα≤˜˜ρα k≤2˜˜ρα 1 2˜ρα≤˜ρα k≤2˜ραand1 2ρα≤ρα k≤2ρα wheneverk≥k(α). Thus for all α,˜˜ρα kand˜˜ραare comparable when kis large enough so we can work with balls of a uniform size, and the same is true for ˜ρα kand ˜ρα, and ρα kandρα. Let ˆBα k=B(xα k,4˜˜ρα), then ˜˜ρα k≤2˜˜ραandB(xα k,2˜˜ρα k)⊂B(xα k,4˜˜ρα) =ˆBα k. So for every rif we letk(r) = max {k(α)|α≤λ(r)}then when k≥k(r), the balls ˆBα kforα≤λ(r) cover the ball B(pk,r) as well. Suppose that ˆBα kandˆBβ kmeet for k≥k(α) andk≥k(β), and suppose rβ k≤rα k. Then, by the triangle inequality, we must have rα k≤rβ k+ 4˜˜ρα+ 4˜˜ρβ≤rβ k+ 8˜˜ρβ<rβ k+ 16˜ρβ k. This then implies ˜˜ρβ k=˜˜ρ(rβ k) = ˜ρ(rβ k+ 20˜ρ(rβ k))<˜ρ(rα k) = ˜ρα k and hence ˜˜ρβ≤4˜ρα. Therefore ˆBβ k⊂B(xα k,36˜ρα) whenever ˆBα kandˆBβ kmeet andk≥max{k(α), k(β)}. Next we deﬁne the balls Bα k=B(xα k,5˜˜ρα) and ˜Bα k=B(xα k,˜˜ρα/2). Note that ˜Bα k are disjoint since ˜Bα k⊂B(xα k,˜˜ρα k). Since ˆBα k⊂Bα k, the ballsBα kcoverB(pk,r) for α≤λ(r) as before. If Bα kandBβ kmeet fork≥k(α) andk≥k(β) andrβ k≤rα k, then by the triangle inequality we get rα k≤rβ k+ 10˜˜ρβ<rβ k+ 20˜ρβ k, and hence ˜˜ρβ≤4˜ρα again. Similarly, ˜ρβ k= ˜ρ(rβ k) =ρ(rβ k+ 20ρ(rβ k))<ρ(rα k) =ρα k. This makes ˜ρβ≤4ρα. Now any point in Bβ khas distance at most 5˜˜ρα+ 5˜˜ρβ+ 5˜˜ρβ≤45˜ρα fromxα k, soBβ k⊂B(xα k,45˜ρα). Likewise, whenever Bα kandBβ kmeet fork≥k(α) andk≥k(β), any point in the larger ball B(xβ k,45˜ρβ) has distance at most 5˜˜ρα+ 5˜˜ρβ+ 45˜ρβ≤205ρα THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 271 fromxα kand henceB(xβ k,45˜ρβ)⊂B(xα k,205ρα). Now we deﬁne ¯Bα k=B(xα k,45˜ρα) and¯¯Bα k=B(xα k,205ρα). Then the above discussion says that whenever Bα kandBβ k meet fork≥k(α) andk≥k(β), we have (4.1.6) Bβ k⊂¯Bα kand ¯Bβ k⊂¯¯Bα k. Note that¯¯Bα kis still a nice embedded ball since, by (4.1.3), 205 ρα≤410ρα k< inj(Mk,xα k). We claim there exist positive numbers N(r) andk′′(r) such that for any given α withrα<r, ask≥k′′(r), there holds (4.1.7) # {β|Bα k∩Bβ k/\e}atio\slash=φ} ≤N(r). Indeed, ifBα kmeetsBβ kthen there is a positive k′′(α) such that as k≥k′′(α), rβ k≤rα k+ 10˜˜ρα k ≤r+ 20ρ(r) ≤r+1 5(s∞−r), where we used (4.1.2) in the third inequality. Set k′′(r) = max {k′′(α),k′(r)|α≤λ(r)} and N(r) =λ/parenleftbigg r+1 5(s∞−r)/parenrightbigg . Then by combining with (4.1.4), these give the desired estim ate (4.1.7) Next we observe that by passing to another subsequence we can guarantee that for any pair αandβwe can ﬁnd a number k(α,β) such that if k≥k(α,β) then either Bα kalways meets Bβ kor it never does. Hence by setting ¯k(r) = max/braceleftbigg k(α,β),k(α),k(β),k′′(r)|α≤λ(r) and β≤λ/parenleftbigg r+1 5(s∞−r)/parenrightbigg/bracerightbigg , we have shown the following results: for every r<s ∞, ifk≥¯k(r), we have (i) the ball B(pk,r) inMkis covered by the balls Bα kforα≤λ(r), (ii) whenever Bα kandBβ kmeet forα≤λ(r), we have Bβ k⊂¯Bα kand ¯Bβ k⊂¯¯Bα k, (iii) for each α≤λ(r), there no more than N(r) balls ever meet Bα k, and (iv) for any α≤λ(r) and anyβ, eitherBα kmeetsBβ kfor allk≥¯k(r) or none for allk≥¯k(r). 272 H.-D. CAO AND X.-P. ZHU Now we let ˆEα,Eα,¯Eα, and¯¯Eαbe the balls of radii 4 ˜˜ρα,5˜˜ρα,45˜ρα, and 205ρα around the origin in Euclidean space Rn. At each point xα k∈B(pk,sk) we deﬁne coordinate charts Hα k:Eα→Bα kas the composition of a linear isometry of Rnto the tangent space Txα kMkwith the exponential map expxα katxα k. We also get maps ¯Hα k:¯Eα→¯Bα kand¯¯Hα k:¯¯Eα→¯¯Bα kin the same way. Note that (4.1.3) implies that these maps are all well deﬁned. We denote by gα k(and ¯gα kand¯¯gα k) the pull-backs of the metric gkbyHα k(and ¯Hα kand¯¯Hα k). We also consider the coordinate transition functionsJαβ k:Eβ→¯Eαand¯Jαβ k:¯Eβ→¯¯Eαdeﬁned by Jαβ k= (¯Hα k)−1Hβ kand ¯Jαβ k= (¯¯Hα k)−1¯Hβ k. Clearly ¯Jαβ kJβα k=I. Moreover Jαβ kis an isometry from gβ kto ¯gα kand¯Jαβ kfrom ¯gβ kto ¯¯gα k. Now for each ﬁxed α, the metrics gα kare in geodesic coordinates and have their curvatures and their covariant derivatives uniformly boun ded. Claim 1. By passing to another subsequence we can guarantee that for e achα (and indeed all αby diagonalization) the metrics gα k(or ¯gα kor¯¯gα k) converge uniformly with their derivatives to a smooth metric gα(or ¯gαor¯¯gα) onEα(or¯Eαor¯¯Eα) which is also in geodesic coordinates. Look now at any pair α,βfor which the balls Bα kandBβ kalways meet for large k, and thus the maps Jαβ k(and ¯Jαβ kandJβα kand¯Jβα k) are always deﬁned for large k. Claim 2. The isometries Jαβ k(and ¯Jαβ kandJβα kand¯Jβα k) always have a conver- gent subsequence. So by passing to another subsequence we may assume Jαβ k→Jαβ(and ¯Jαβ k→ ¯JαβandJβα k→Jβαand¯Jβα k→¯Jβα). The limit maps Jαβ:Eβ→¯Eαand ¯Jαβ:¯Eβ→¯¯Eαare isometries in the limit metrics gβandgα. Moreover Jαβ¯Jβα=I. We are now done picking subsequences, except we still owe the reader the proofs of Claim 1 and Claim 2. Step2: Finding local diﬀeomorphisms which are approximate isom etries. Take the subsequence ( B(pk,sk),gk,pk) chosen in Step 1 above. We claim that for everyr < s ∞and every ( ǫ1,ǫ2,...,ǫ p), and for all kandlsuﬃciently large in comparison, we can ﬁnd a diﬀeomorphism Fklof a neighborhood of the ball B(pk,r)⊂ B(pk,sk) into an open set in B(pl,sl) which is an ( ǫ1,ǫ2,...,ǫ p) approximate isometry in the sense that |t∇Fkl∇Fkl−I|<ǫ1 and |∇2Fkl|<ǫ2,...,|∇pFkl|<ǫp where ∇pFklis thepthcovariant derivative of Fkl. The idea (following Peters [106] or Greene and Wu [51]) of pro ving the claim is to deﬁne the map Fα kℓ=Hα l◦(Hα k)−1(or¯Fα kℓ=¯Hα l◦(¯Hα k)−1, resp.) from Bα ktoBα ℓ THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 273 (or¯Bα kto¯Bα ℓ, resp.) for kandℓlarge compared to αso as to be the identity map onEα(or¯Eα, resp.) in the coordinate charts Hα kandHα ℓ(or¯Hα kand¯Hα ℓ, resp.), and then to deﬁne Fkℓon a neighborhood of B(pk,r) fork,ℓ≥¯k(r) be averaging the maps ¯Fβ kℓforβ≤λ(r+1 5(s∞−r)). To describe the averaging process on Bα k withα≤λ(r) we only need to consider those Bβ kwhich meet Bα k; there are never more thanN(r) of them and each β≤λ(r+1 5(s∞−r)), and they are the same for k andℓwhenk,ℓ≥¯k(r). The averaging process is deﬁned by taking Fkℓ(x) to be the center of mass of the ¯Fβ kℓ(x) forx∈Bα kaveraging over those βwhereBβ kmeetsBα k using weights µβ k(x) deﬁned by a partition of unity. The center of mass of the poin ts yβ=Fβ kℓ(x) with weights µβis deﬁned to be the point ysuch that expyVβ=yβand/summationdisplay µβVβ= 0. When the points yβare all close and the weights µβsatisfy 0 ≤µβ≤1 then there will be a unique solution yclose toyβwhich depends smoothly on the yβand theµβ (see [51] for the details). The point yis found by the inverse function theorem, which also provides bounds on all the derivatives of yas a function of the yβand theµβ. SinceBα k⊆¯Bβ kand¯Bβ ℓ⊆¯¯Bα ℓ, the map ¯Fβ kℓ=¯Hβ l◦(¯Hβ k)−1can be represented in local coordinates by the map Pαβ kℓ:Eα→¯¯Eα deﬁned by Pαβ kℓ=¯Jαβ ℓ◦Jβα k. SinceJβα k→Jβαask→ ∞ and¯Jαβ ℓ→¯Jαβasℓ→ ∞ and¯Jαβ◦Jβα=I, we see that the maps Pαβ kℓ→Iask,ℓ→ ∞ for each choice of αandβ. The weights µβ kare deﬁned in the following way. We pick for each βa smooth function ψβwhich equals 1 on ˆEβand equals 0 outside Eβ. We then transfer ψβto a function ψβ konMkby the coordinate map¯¯Hβ k(i.e.ψβ k=ψβ◦(¯¯Hβ k)−1). Then let µβ k=ψβ k/slashig/summationdisplay γψγ k as usual. In the coordinate chart Eαthe function ψβ klooks like the composition of Jβα kwithψβ. Call this function ψαβ k=ψβ◦Jβα k. Then ask→ ∞,ψαβ k→ψαβwhere ψαβ=ψβ◦Jβα. In the coordinate chart Eαthe function µβ klooks like µαβ k=ψαβ k/slashig/summationdisplay γψαγ k andµαβ k→µαβask→ ∞ where µαβ=ψαβ/slashig/summationdisplay γψαγ. 274 H.-D. CAO AND X.-P. ZHU Since the sets ˆBα kcoverB(pk,r), it follows that/summationtext γψγ k≥1 on this set and by combining with (4.1.5) and (4.1.7) there is no problem bound ing all these functions and their derivatives. There is a small problem in that we wan t to guarantee that the averaged map still takes pktopℓ. This is true at least for the map F0 kℓ. Therefore it will suﬃce to guarantee that µα k= 0 in a neighborhood of pkifα/\e}atio\slash= 0. This happens if the same is true for ψα k. If not, we can always replace ψα kby˜ψα k= (1−ψ0 k)ψα kwhich still leaves ˜ψα k≥1 2ψα korψ0 k≥1 2everywhere, and this is suﬃcient to make/summationtext γ˜ψγ k≥1 2 everywhere. Now in the local coordinate Eαwe are averaging maps Pαβ kℓwhich converge to the identity with respect to weights µαβ kwhich converge. It follows that the averaged map converges to the identity in these coordinates. Thus Fkℓcan be made to be an (ǫ1,ǫ2,...,ǫ p) approximate isometry on B(pk,r) whenkandℓare suitably large. At least the estimates |t∇Fkℓ· ∇Fkℓ−I|<ǫ1 and|∇2Fkℓ|<ǫ2,...,|∇pFkℓ|<ǫponB(pk,r) follow from the local coordinates. We still need to check that Fkℓis a diﬀeomorphism on a neighborhood of B(pk,r). This, however, follows quickly enough from the fact that we a lso get a map Fℓk on a slightly larger ball B(pℓ,r′) which contains the image of FkℓonB(pk,r) if we taker′= (1 +ǫ1)r, andFℓkalso satisﬁes the above estimates. Also FkℓandFℓkﬁx the markings, so the composition Fℓk◦Fkℓsatisﬁes the same sort of estimates and ﬁxes the origin pk. Since the maps Pαβ kℓandPαβ ℓkconverge to the identity as k,ℓtend to inﬁnity, Fℓk◦Fkℓmust be very close to the identity on B(pk,r). It follows that Fkℓis invertible. This ﬁnishes the proof of the claim and the Step 2. Step3: Constructing the limit geodesic ball ( B∞,g∞,p∞). We now know the geodesic balls ( B(pk,sk),gk,pk) are nearly isometric for large k. We are now going to construct the limit B∞. For a sequence of positive numbers rjրs∞with eachrj<sj, we choose the numbers ( ǫ1(rj),...,ǫ j(rj)) so small that when we choose k(rj) large in comparison and ﬁnd the maps Fk(rj),k(rj+1)constructed above on neighborhoods of B(pk(rj),rj), inMk(rj)intoMk(rj+1)the image always lies inB(pk(rj+1),rj+1) and the composition of Fk(rj),k(rj+1)withFk(rj+1),k(rj+2)and··· andFk(rs−1),k(rs)for anys>j is still an (η1(rj),...,η j(rj)) isometry for any choice ofηi(rj), sayηi(rj) = 1/jfor 1≤i≤j. Now we simplify the notation by writing Mj in place ofMk(rj)andFjin place ofFk(rj),k(rj+1). Then Fj:B(pj,rj)→B(pj+1,rj+1) is a diﬀeomorphism map from B(pj,rj) intoB(pj+1,rj+1), and the composition Fs−1◦ ··· ◦Fj:B(pj,j)→B(ps,s) is always an ( η1(rj),...,η j(rj)) approximate isometry. We now construct the limit B∞as a topological space by identifying the balls B(pj,rj) with each other using the homeomorphisms Fj. Given any two points xand yinB∞, we havex∈B(pj,rj) andy∈B(ps,rs) for some jands. Ifj≤sthen x∈B(ps,rs) also, by identiﬁcation. A set in B∞is open if and only if it intersects eachB(pj,rj) in an open set. Then choosing disjoint neighborhoods of xandyin THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 275 B(ps,rs) gives disjoint neighborhoods of xandyinB∞. ThusB∞is a Hausdorﬀ space. Any smooth chart on B(pj,rj) also gives a smooth chart on B(ps,rs) for alls>j. The union of all such charts gives a smooth atlas on B∞. It is fairly easy to see the metricsgjonB(pj,rj), converge to a smooth metric g∞onB∞uniformly together with all derivatives on compact sets. For since the Fs−1◦ ··· ◦Fjare very good approximate isometries, the gjare very close to each other, and hence form a Cauchy sequence (together with their derivatives, in the sense tha t the covariant derivatives ofgjwith respect to gsare very small when jandsare both large). One checks in the usual way that such a Cauchy sequence converges. The origins pjare identiﬁed with each other, and hence with an origin p∞inB∞. Now it is the inverses of the maps identifying B(pj,rj) with open subsets of B∞that provide the diﬀeomorphisms of (relatively compact) open se ts inB∞into the geodesic ballsB(pj,sj)⊂Mjsuch that the pull-backs of the metrics gjconverge to g∞. This completes the proof of Step 3. Now it remains to prove both Claim 1 and Claim 2 in Step 1. Proof of Claim 1. It suﬃces to show the following general result: There exists a constant c >0depending only on the dimension, and constants Cqdepending only on the dimension and qand bounds Bjon the curvature and its derivatives for j≤qwhere |DjRm| ≤Bj, so that for any metric gkℓin geodesic coordinates in the ball |x| ≤r≤c/√B0, we have 1 2Ikℓ≤gkℓ≤2Ikℓ and /vextendsingle/vextendsingle/vextendsingle∂ ∂xj1···∂ ∂xjqgkℓ/vextendsingle/vextendsingle/vextendsingle≤Cq, whereIkℓis the Euclidean metric. Suppose we are given a metric gij(x)dxidxjin geodesic coordinates in the ball |x| ≤r≤c/√B0as in Claim 1. Then by deﬁnition every line through the origin is a geodesic (parametrized proportional to arc length) and gij=Iijat the origin. Also, the Gauss Lemma says that the metric gijis in geodesic coordinates if and only if gijxi=Iijxi. Note in particular that in geodesic coordinates |x|2=gijxixj=Iijxixj is unambiguously deﬁned. Also, in geodesic coordinates we h ave Γk ij(0) = 0, and all the ﬁrst derivatives for gjkvanish at the origin. Introduce the symmetric tensor Aij=1 2xk∂ ∂xkgij. Since we have gjkxk=Ijkxk, we get xk∂ ∂xigjk=Iij−gij=xk∂ ∂xjgik 276 H.-D. CAO AND X.-P. ZHU and hence from the formula for Γi jk xjΓi jk=giℓAkℓ. HenceAkℓxk= 0. LetDibe the covariant derivative with respect to the metric gij. Then Dixk=Ik i+ Γk ijxj=Ik i+gkℓAiℓ. Introduce the potential function P=|x|2/2 =1 2gijxixj. We can use the formulas above to compute DiP=gijxj. Also we get DiDjP=gij+Aij. The deﬁning equation for Pgives gijDiPDjP= 2P. If we take the covariant derivative of this equation we get gkℓDjDkPDℓP=DjP which is equivalent to Ajkxk= 0. But if we take the covariant derivative again we get gkℓDiDjDkPDℓP+gkℓDjDkPDiDℓP=DiDjP. Now switching derivatives DiDjDkP=DiDkDjP=DkDiDjP+RikjℓgℓmDmP and if we use this and DiDjP=gij+AijandgkℓDℓP=xkwe ﬁnd that xkDkAij+Aij+gkℓAikAjℓ+Rikjℓxkxℓ= 0. From our assumed curvature bounds we can take |Rijkℓ| ≤B0. Then we get the following estimate: |xkDkAij+Aij| ≤C|Aij|2+CB0r2 on the ball |x| ≤rfor some constant Cdepending only on the dimension. We now show how to use the maximum principle on such equations . First of all, by a maximum principle argument, it is easy to show that if fis a function on a ball |x| ≤randλ>0 is a constant, then λsup|f| ≤sup/vextendsingle/vextendsingle/vextendsinglexk∂f ∂xk+λf/vextendsingle/vextendsingle/vextendsingle. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 277 For any tensor T={Ti···j}and any constant λ >0, settingf=|T|2in the above inequality, we have (4.1.8) λsup|T| ≤sup|xkDkT+λT|. Applying this to the tensor Aijwe get sup |x|≤r|Aij| ≤Csup |x|≤r|Aij|2+CB0r2 for some constant depending only on the dimension. It is fairly elementary to see that there exist constants c >0 andC0<∞such that if the metric gijis in geodesic coordinates with |Rijkℓ| ≤B0in the ball of radius r≤c/√B0then |Aij| ≤C0B0r2. Indeed, since the derivatives of gijvanish at the origin, so does Aij. Hence the estimate holds near the origin. But the inequality sup |x|≤r|Aij| ≤Csup |x|≤r|Aij|2+CB0r2 says that |Aij|avoids an interval when cis chosen small. In fact the inequality X≤CX2+D is equivalent to |2CX−1| ≥√ 1−4CD which makes Xavoid an interval if 4 CD < 1. (Hence in our case we need to choose cwith 4C2c2<1.) Then ifXis on the side containing 0 we get X≤1−√ 1−4CD 2C≤2D. This gives |Aij| ≤C0B0r2withC0= 2C. We can also derive bounds on all the covariant derivatives of Pin terms of bounds on the covariant derivatives of the curvature. To simplify t he notation, we let DqP={Dj1Dj2···DjqP} denote the qthcovariant derivative, and in estimating DqPwe will lump all the lower order terms into a general slush term Φqwhich will be a polynomial in D1P,D2P,...,Dq−1PandRm,D1Rm,...,Dq−2Rm. We already have estimates on a ball of radius r P≤r2/2 |D1P| ≤r |Aij| ≤C0B0r2 278 H.-D. CAO AND X.-P. ZHU and sinceDiDjP=gij+Aijandr≤c/√B0if we choose csmall we can make |Aij| ≤1/2, and we get |D2P| ≤C2 for some constant C2depending only on the dimension. Start with the equation gijDiPDjP= 2Pand apply repeated covariant deriva- tives. Observe that we get an equation which starts out gijDiPDqDjP+···= 0 where the omitted terms only contain derivatives DqPand lower. If we switch two derivatives in a term Dq+1Por lower, we get a term which is a product of a covariant derivative of Rmof order at most q−2 (since the two closest to Pcommute) and a covariant derivative of Pof order at most q−1; such a term can be lumped in with the slush term Φq. Therefore up to terms in Φqwe can regard the derivatives as commuting. Then paying attention to the derivatives in D1Pwe get an equation gijDiPDjDk1···DkqP+gijDiDk1PDjDk2···DkqP +gijDiDk2PDjDk1Dk3···DkqP+···+gijDiDkqPDjDk1···Dkq−1P =Dk1···DkqP+ Φq. Recalling that DiDjP=gij+Aijwe can rewrite this as Φq=gijDiPDjDk1···Dkq+ (q−1)Dk1···DkqP +gijAik1DjDk2···DkqP+···+gijAikqDjDk1···Dkq−1P. Estimating the product of tensors in the usual way gives |xiDiDqP+ (q−1)DqP| ≤q|A||DqP|+|Φq|. Applying the inequality λsup|T| ≤sup|xkDkT+λT|withT=DqPgives (q−1)sup |DqP| ≤sup(q|A||DqP|+|Φq|). Now we can make |A| ≤1/2 by making r≤c/√B0withcsmall; it is important here thatcis independent of q! Then we get (q−2)sup |DqP| ≤2 sup|Φq| which is a good estimate for q≥3. The term Φqis estimated inductively from the termsDq−1PandDq−2Rmand lower. This proves that there exist constants Cqfor q≥3 depending only on qand the dimension and on |DjRm|forj≤q−2 such that |DqP| ≤Cq on the ball r≤c/√B0. Now we turn our attention to estimating the Euclidean metric Ijkand its covariant derivatives with respect to gjk. We will need the following elementary fact: suppose thatfis a function on a ball |x| ≤rwithf(0) = 0 and /vextendsingle/vextendsingle/vextendsinglexi∂f ∂xi/vextendsingle/vextendsingle/vextendsingle≤C|x|2 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 279 for some constant C. Then (4.1.9) |f| ≤C|x|2 for the same constant C. As a consequence, if T={Tj···k}is a tensor which vanishes at the origin and if |xiDiT| ≤C|x|2 on a ball |x| ≤rthen|T| ≤C|x|2with the same constant C. (Simply apply the inequality (4.1.9) to the function f=|T|. In case this is not smooth, we can use f=/radicalbig |T|2+ǫ2−ǫand then let ǫ→0.) Our application will be to the tensor Ijkwhich gives the Euclidean metric as a tensor in geodesic coordinates. We have DiIjk=−Γp ijIpk−Γp ikIpj and since xiΓp ij=gpqAjq we get the equation xiDiIjk=−gpqAjpIkq−gpqAkpIjq. We already have |Ajk| ≤C0B0|x|2for|x| ≤r≤c/√B0. The tensor Ijkdoesn’t vanish at the origin, but the tensor hjk=Ijk−gjk does. We can then use xiDihjk=−gpqAjphkq−gpqAkqhjq−2Ajk. SupposeM(s) = sup|x|≤s|hjk|. Then |xiDihjk| ≤2[1 +M(s)]C0B0|x|2 and we get |hjk| ≤2[1 +M(s)]C0B0|x|2 on|x| ≤s. This makes M(s)≤2[1 +M(s)]C0B0s2. Then fors≤r≤c/√B0withcsmall compared to C0we get 2C0B0s2≤1/2 and M(s)≤4C0B0s2. Thus |Ijk−gjk|=|hjk| ≤4C0B0|x|2 for|x| ≤r≤c/√B0, and hence for csmall enough 1 2gjk≤Ijk≤2gjk. 280 H.-D. CAO AND X.-P. ZHU Thus the metrics are comparable. Note that this estimate onl y needsrsmall compared toB0and does not need any bounds on the derivatives of the curvatu re. Now to obtain bounds on the covariant derivative of the Eucli den metric Ikℓwith respect to the Riemannian metric gkℓwe want to start with the equation xiDiIkℓ+gmnAkmIℓn+gmnAℓmIkn= 0 and applyqcovariant derivatives Dj1···Djq. Each time we do this we must inter- changeDjandxiDi, and since this produces a term which helps we should look at it closely. If we write Rji= [Dj,Di] for the commutator, this operator on tensors involves the curvature but no derivatives. Since Djxi=Ii j+gimAjm we can compute [Dj,xiDi] =Dj+gimAjmDi+xiRji and the term Djin the commutator helps, while Ajmcan be kept small and Rjiis zero order. It follows that we get an equation of the form 0 =xiDiDj1···DjqIkℓ+qDj1···DjqIkℓ +q/summationdisplay h=1gimAjhmDj1···Djh−1DiDjh+1···DjqIkℓ +gmnAkmDj1···DjqIℓn+gmnAℓmDj1···DjqIkn+ Ψq, where the slush term Ψqis a polynomial in derivatives of Ikℓof degree no more than q−1 and derivatives of Pof degree no more than q+ 2 (remember xi=gijDjPand Aij=DiDjP−gij) and derivatives of the curvature Rmof degree no more than q−1. We now estimate DqIkℓ={Dj1···DjqIkℓ} by induction on qusing (4.1.8) with λ=q. Noticing a total of q+2 terms contracting Aijwith a derivative of Ikℓof degreeq, we get the estimate qsup|DqIkℓ| ≤(q+ 2)sup |A|sup|DqIkℓ|+ sup|Ψq|. and everything works. This proves that there exists a consta ntc >0 depending only on the dimension, and constants Cqdepending only on the dimension and qand boundsBjon the curvature and its derivatives for j≤qwhere |DjRm| ≤Bj, so that for any metric gkℓin geodesic coordinates in the ball |x| ≤r≤c/√B0the Euclidean metricIkℓsatisﬁes 1 2gkℓ≤Ikℓ≤2gkℓ and the covariant derivatives of Ikℓwith respect to gkℓsatisfy |Dj1···DjqIkℓ| ≤Cq. The diﬀerence between a covariant derivative and an ordinar y derivative is given by the connection −Γp ijIpk−Γp ikIpj THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 281 to get Γk ij=1 2Ikℓ(DℓIij−DiIjℓ−DjIiℓ). This gives us bounds on Γk ij. We then obtain bounds on the ﬁrst derivatives of gij from ∂ ∂xigjk=gkℓΓℓ ij+gjℓΓℓ ik. Always proceeding inductively on the order of the derivativ e, we now get bounds on covariant derivatives of Γk ijfrom the covariant derivatives of Ipkand bounds of the ordinary derivatives of Γk ijby relating the to the covariant derivatives using the Γk ij, and bounds on the ordinary derivatives of the gjkfrom bounds on the ordinary derivatives of the Γℓ ij. Consequently, we have estimates 1 2Ikℓ≤gkℓ≤2Ikℓ and /vextendsingle/vextendsingle/vextendsingle∂ ∂xj1···∂ ∂xjqgkℓ/vextendsingle/vextendsingle/vextendsingle≤˜Cq for similar constants ˜Cq. Therefore we have ﬁnished the proof of Claim 1. Proof of Claim 2. We need to show how to estimate the derivatives of an isometry . We will prove that if y=F(x) is an isometry from a ball in Euclidean space with a metricgijdxidxjto a ball in Euclidean space with a metric hkldykdyl. Then we can bound all of the derivatives of ywith respect to xin terms of bounds on gijand its derivatives with respect to xand bound on hkland its derivatives with respect to y. This would imply Claim 2. Sincey=F(x) is an isometry we have the equation hpq∂yp ∂xj∂yq ∂xk=gjk. Using bounds gjk≤CIjkandhpq≥cIpqcomparing to the Euclidean metric, we easily get estimates /vextendsingle/vextendsingle/vextendsingle∂yp ∂xj/vextendsingle/vextendsingle/vextendsingle≤C. Now if we diﬀerentiate the equation with respect to xiwe get hpq∂2yp ∂xi∂xj∂yq ∂xk+hpq∂yp ∂xj∂2yq ∂xi∂xk=∂gjk ∂xi−∂hpq ∂yr∂yr ∂xi∂yp ∂xj∂yq ∂xk. Now let Tijk=hpq∂yp ∂xi∂2yq ∂xj∂xk 282 H.-D. CAO AND X.-P. ZHU and let Uijk=∂gjk ∂xi−∂hpq ∂yr∂yr ∂xi∂yp ∂xj∂yq ∂xk. Then the above equation says Tkij+Tjik=Uijk. Using the obvious symmetries Tijk=TikjandUijk=Uikjwe can solve this in the usual way to obtain Tijk=1 2(Ujik+Ukij−Uijk). We can recover the second derivatives of ywith respect to xfrom the formula ∂2yp ∂xi∂xj=gkℓTkij∂yp ∂xℓ. Combining these gives an explicit formula giving ∂2yp/∂xi∂xjas a function of gij,hpq,∂gjk/∂xi,∂hpq/∂yr, and∂yp/∂yi. This gives bounds /vextendsingle/vextendsingle/vextendsingle∂2yp ∂yi∂yj/vextendsingle/vextendsingle/vextendsingle≤C and bounds on all higher derivatives follow by diﬀerentiati ng the formula and using induction. This completes the proof of Claim 2 and hence the p roof of Theorem 4.1.2. We now want to show how to use this convergence result on solut ions to the Ricci ﬂow. Let us ﬁrst state the deﬁnition for the convergence of ev olving manifolds. Definition 4.1.3. Let (Mk,gk(t),pk) be a sequence of evolving marked com- plete Riemannian manifolds, with the evolving metrics gk(t) over a ﬁxed time in- tervalt∈(A,Ω],A <0≤Ω, and with the marked points pk∈Mk. We say a sequence of evolving marked ( B0(pk,sk),gk(t),pk) overt∈(A,Ω], whereB0(pk,sk) are geodesic balls of ( Mk,gk(0)) centered at pkwith the radii sk→s∞(≤+∞),con- verges in theC∞ loctopology to an evolving marked (maybe noncomplete) manifold (B∞,g∞(t),p∞) overt∈(A,Ω], where, at the time t= 0,B∞is a geodesic open ball centered at p∞∈B∞with the radius s∞, if we can ﬁnd a sequence of exhausting open setsUkinB∞containing p∞and a sequence of diﬀeomorphisms fkof the sets Uk inB∞to open sets VkinB(pk,sk)⊂Mkmappingp∞topksuch that the pull-back metrics ˜gk(t) = (fk)∗gk(t) converge in C∞topology to g∞(t) on every compact subset ofB∞×(A,Ω]. Now we ﬁx a time interval A < t ≤Ω with −∞< A < 0 and 0 ≤Ω<+∞. Consider a sequence of marked evolving complete manifolds ( Mk,gk(t),pk), t∈(A,Ω], with eachgk(t),k= 1,2,...,being a solution of the Ricci ﬂow ∂ ∂tgk(t) =−2Ric k(t) onB0(pk,sk)×(A,Ω], whereRickis the Ricci curvature tensor of gk, andB0(pk,sk) is the geodesic ball of ( Mk,gk(0)) centered at pkwith the radii sk→s∞(≤+∞). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 283 Assume that for each r < s ∞there are positive constants C(r) andk(r) such that the curvatures of gk(t) satisfy the bound |Rm(gk)| ≤C(r) onB0(pk,r)×(A,Ω] for allk≥k(r).We also assume that ( Mk,gk(t),pk),k= 1,2,..., have a uniform injectivity radius bound at the origins pkatt= 0. By Shi’s derivatives estimate (Theorem 1.4.1), the above assumption of uniform b ound of the curvatures on the geodesic balls B0(pk,r) (r<s ∞) implies the uniform bounds on all the deriv- atives of the curvatures at t= 0 on the geodesic balls B0(pk,r) (r < s ∞). Then by Theorem 4.1.2 we can ﬁnd a subsequence of marked evolving man ifolds, still denoted by (Mk,gk(t),pk) witht∈(A,Ω], so that the geodesic balls ( B0(pk,sk),gk(0),pk) converge in the C∞ loctopology to a geodesic ball ( B∞(p∞,s∞),g∞(0),p∞). From now on, we consider this subsequence of marked evolving manifol ds. By Deﬁnition 4.1.1, we have a sequence of (relatively compact) exhausting cover ing{Uk}ofB∞(p∞,s∞) containingp∞and a sequence of diﬀeomorphisms fkof the setsUkinB∞(p∞,s∞) to open setsVkinB0(pk,sk) mappingp∞topksuch that the pull-back metrics at t= 0 ˜gk(0) = (fk)∗gk(0)C∞ loc−→g∞(0),ask→+∞,onB∞(p∞,s∞). However, the pull-back metrics ˜ gk(t) = (fk)∗gk(t) are also deﬁned at all times A < t≤Ω (although g∞(t) is not yet). We also have uniform bounds on the curvature of the pull-back metrics ˜ gk(t) and all their derivatives, by Shi’s derivative estimates (Theorem 1.4.1), on every compact subset of B∞(p∞,s∞)×(A,Ω]. What we claim next is that we can ﬁnd uniform bounds on all the covariant der ivatives of the ˜ gk taken with respect to the ﬁxed metric g∞(0). Lemma 4.1.4. Let(M,g)be a Riemannian manifold, Ka compact subset of M, and˜gk(t)a collection of solutions to Ricci ﬂow deﬁned on neighborhoo ds ofK×[α,β] with[α,β]containing 0. Suppose that for each l≥0, (a)C−1 0g≤˜gk(0)≤C0g, on K, for all k, (b)|∇l˜gk(0)| ≤Cl, on K, for all k, (c)|˜∇l kRm(˜gk)|k≤C′ l, on K ×[α,β], for all k, for some positive constants Cl, C′ l, l= 0,1,...,independent of k, whereRm(˜gk)are the curvature tensors of the metrics ˜gk(t),˜∇kdenote covariant derivative with respect to˜gk(t),|·|kare the length of a tensor with respect to ˜gk(t), and|·|is the length with respect tog. Then the metrics ˜gk(t)satisfy ˜C0−1g≤˜gk(t)≤˜C0g, on K ×[α,β] and |∇l˜gk| ≤˜Cl, on K ×[α,β], l= 1,2,..., for allk, where ˜Cl, l= 0,1,...,are positive constants independent of k. Proof. First by using the equation ∂ ∂t˜gk=−2˜Rick and the assumption (c) we immediately get (4.1.10) ˜C0−1g≤˜gk(t)≤˜C0g,onK×[α,β] 284 H.-D. CAO AND X.-P. ZHU for some positive constant ˜C0independent of k. Next we want to bound ∇˜gk. The diﬀerence of the connection ˜Γkof ˜gkand the connection Γ of gis a tensor. Taking Γ to be ﬁxed in time, we get ∂ ∂t(˜Γk−Γ) =∂ ∂t/parenleftbigg1 2(˜gk)γδ/bracketleftbigg∂ ∂xα(˜gk)δβ+∂ ∂xβ(˜gk)δα−∂ ∂xδ(˜gk)αβ/bracketrightbigg/parenrightbigg =1 2(˜gk)γδ/bracketleftig (˜∇k)α(−2(˜Rick)βδ) + (˜∇k)β(−2(˜Rick)αδ) −(˜∇k)δ(−2(˜Rick)αβ)/bracketrightig and then by the assumption (c) and (4.1.10), /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ ∂t(˜Γk−Γ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C,for allk. Note also that at a normal coordinate of the metric gat a ﬁxed point and at the time t= 0, (˜Γk)γ αβ−Γγ αβ=1 2(˜gk)γδ/parenleftbigg∂ ∂xα(˜gk)δβ+∂ ∂xβ(˜gk)δα−∂ ∂xδ(˜gk)αβ/parenrightbigg (4.1.11) =1 2(˜gk)γδ(∇α(˜gk)δβ+∇β(˜gk)δα− ∇ δ(˜gk)αβ), thus by the assumption (b) and (4.1.10), |˜Γk(0)−Γ| ≤C,for allk. Integrating over time we deduce that (4.1.12) |˜Γk−Γ| ≤C,onK×[α,β],for allk. By using the assumption (c) and (4.1.10) again, we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ ∂t(∇˜gk)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=| −2∇˜Rick| =| −2˜∇k˜Rick+ (˜Γk−Γ)∗˜Rick| ≤C,for allk. Hence by combining with the assumption (b) we get bounds (4.1.13) |∇˜gk| ≤˜C1,onK×[α,β], for some positive constant ˜C1independent of k. Further we want to bound ∇2˜gk. Again regarding ∇as ﬁxed in time, we see ∂ ∂t(∇2˜gk) =−2∇2(˜Rick). Write ∇2˜Rick= (∇ −˜∇k)(∇˜Rick) +˜∇k(∇ −˜∇k)˜Rick+˜∇2 k˜Rick = (Γ−˜Γk)∗ ∇˜Rick+˜∇k((Γ−˜Γk)∗˜Rick) +˜∇2 k˜Rick = (Γ−˜Γk)∗[(∇ −˜∇k)˜Rick+˜∇k˜Rick] +˜∇k(˜g−1 k∗ ∇˜gk∗˜Rick) +˜∇2 k˜Rick = (Γ−˜Γk)∗[(Γ−˜Γk)∗˜Rick+˜∇k˜Rick] +˜∇k(˜g−1 k∗ ∇˜gk∗˜Rick) +˜∇2 k˜Rick THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 285 where we have used (4.1.11). Then by the assumption (c), (4.1 .10), (4.1.12) and (4.1.13) we have |∂ ∂t∇2˜gk| ≤C+C· |˜∇k∇˜gk| =C+C· |∇2˜gk+ (˜Γk−Γ)∗ ∇˜gk| ≤C+C|∇2˜gk|. Hence by combining with the assumption (b) we get |∇2˜gk| ≤˜C2,onK×[α,β], for some positive constant ˜C2independent of k. The bounds on the higher derivatives can be derived by the sam e argument. Therefore we have completed the proof of the lemma. We now apply the lemma to the pull-back metrics ˜ gk(t) = (fk)∗gk(t) on B∞(p∞,s∞)×(A,Ω]. Since the metrics ˜ gk(0) have uniform bounds on their cur- vature and all derivatives of their curvature on every compa ct set ofB∞(p∞,s∞) and converge to the metric g∞(0) inC∞ loctopology, the assumptions (a) and (b) are certainly held for every compact subset K⊂B∞(p∞,s∞) withg=g∞(0). For every compact subinterval [ α,β]⊂(A,Ω], we have already seen from Shi’s derivative esti- mates (Theorem 1.4.1) that the assumption (c) is also held on K×[α,β]. Then all of the∇l˜gkare uniformly bounded with respect to the ﬁxed metric g=g∞(0) on every compact set of B∞(p∞,s∞)×(A,Ω]. By using the classical Arzela-Ascoli theorem, we can ﬁnd a subsequence which converges uniformly together with all its derivatives on every compact subset of B∞(p∞,s∞)×(A,Ω]. The limit metric will agree with that obtained previously at t= 0, where we know its convergence already. The limit g∞(t), t∈(A,Ω], is now clearly itself a solution of the Ricci ﬂow. Thus we o btain the following Cheeger type compactness theorem to the Ricci ﬂow , which is essentially obtained by Hamilton in [62] and is called Hamilton’s compactness theorem . Theorem 4.1.5 ( Hamilton’s compactness theorem ).Let(Mk,gk(t),pk), t∈ (A,Ω]withA < 0≤Ω, be a sequence of evolving marked complete Riemannian manifolds. Consider a sequence of geodesic balls B0(pk,sk)⊂Mkof radiisk(0< sk≤+∞), withsk→s∞(≤+∞), around the base points pkin the metrics gk(0). Suppose each gk(t)is a solution to the Ricci ﬂow on B0(pk,sk)×(A,Ω]. Suppose also (i) for every radius r < s ∞there exist positive constants C(r)andk(r)inde- pendent of ksuch that the curvature tensors Rm(gk)of the evolving metrics gk(t)satisfy the bound |Rm(gk)| ≤C(r), onB0(pk,r)×(A,Ω]for allk≥k(r), and (ii) there exists a constant δ>0such that the injectivity radii of Mkatpkin the metricgk(0)satisfy the bound inj(Mk,pk,gk(0))≥δ>0, for allk= 1,2,.... Then there exists a subsequence of evolving marked (B0(pk,sk),gk(t),pk)overt∈ (A,Ω]which converge in C∞ loctopology to a solution (B∞,g∞(t),p∞)overt∈(A,Ω] to the Ricci ﬂow, where, at the time t= 0,B∞is a geodesic open ball centered at p∞∈B∞with the radius s∞. Moreover the limiting solution is complete if s∞= +∞. 286 H.-D. CAO AND X.-P. ZHU 4.2. Injectivity Radius Estimates. We will use rescaling arguments to un- derstand the formation of singularities and long-time beha viors of the Ricci ﬂow. In view of the compactness property obtained in the previous se ction, on one hand one needs to control the bounds on the curvature, and on the other hand one needs to control the lower bounds of the injectivity radius. In appli cations we usually rescale the solution so that the (rescaled) curvatures become unifo rmly bounded on compact subsets and leave the injectivity radii of the (rescaled) so lutions to be estimated in terms of curvatures. In this section we will review a number o f such injectivity ra- dius estimates in Riemannian geometry. In the end we will com bine these injectivity estimates with Perelman’s no local collapsing theorem I′to give the well-known little loop lemma to the Ricci ﬂow which was conjectured by Hamilton in [63]. LetMbe a Riemannian manifold. Recall that the injectivity radius at a point p∈Mis deﬁned by inj (M,p) = sup {r>0|expp:B(O,r)(⊂TpM)→Mis injective }, and the injectivity radius of M is inj (M) = inf {inj(M,p)|p∈M}. We begin with a basic lemma due to Klingenberg (see for exampl e, Corollary 5.7 in Cheeger & Ebin [22]). Klingenberg’s Lemma. LetMbe a complete Riemannian manifold and let p∈M. LetlM(p)denote the minimal length of a nontrivial geodesic loop star ting and ending at p(maybe not smooth at p). Then the injectivity radius of Matp satisﬁes the inequality inj(M,p)≥min/braceleftbiggπ√Kmax,1 2lM(p)/bracerightbigg whereKmaxdenotes the supermum of the sectional curvature on Mand we understand π/√Kmaxto be positive inﬁnity if Kmax≤0. Based on this lemma and a second variation argument, Klingen berg proved that the injectivity radius of an even-dimensional, compact, si mply connected Riemannian manifold of positive sectional curvature is bounded from be low byπ/√Kmax. For odd- dimensional, compact, simply connected Riemannian manifo ld of positive sectional curvature, the same injectivity radius estimates was also p roved by Klingenberg under an additional assumption that the sectional curvature is st rictly1 4-pinched (see for example Theorem 5.9 and 5.10 in Cheeger & Ebin [22]). We also r emark that in dimension 7, there exists a sequence of simply connected, ho mogeneous Einstein spaces whose sectional curvatures are positive and uniformly boun ded from above but their injectivity radii converge to zero. (See [2].) The next result due to Gromoll and Meyer [52] shows that for co mplete, non- compact Riemannian manifold with positive sectional curva ture, the above injectivity radius estimate actually holds without any restriction on d imension. Since the result and proof were not explicitly given in [52], we include a proo f here. Theorem 4.2.1 ( The Gromoll-Meyer injectivity radius estimate ).LetMbe a complete, noncompact Riemannian manifold with positive se ctional curvature. Then the injectivity radius of Msatisﬁes the following estimate inj (M)≥π√Kmax. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 287 Proof. LetObe an arbitrary ﬁxed point in M. We need to show that the injectivity radius at Ois not less than π/√Kmax. We argue by contradiction. Suppose not, then by Klingenberg’s lemma there exists a closed geode sic loopγonMstarting and ending at O(may be not smooth at O). SinceMhas positive sectional curvature, we know from the work of Gr omoll- Meyer [52] (see also Proposition 8.5 in Cheeger & Ebin [22]) t hat there exists a compact totally convex subset CofMcontaining the geodesic loop γ. Among all geodesic loops starting and ending at the same point and lyin g entirely in the compact totally convex set Cthere will be a shortest one. Call it γ0, and suppose γ0starts and ends at a point we call p0. First we claim that γ0must be also smooth at the point p0. Indeed by the curvature bound and implicit function theorem, there will b e a geodesic loop ˜ γclose toγ0starting and ending at any point ˜ pclose top0. Let ˜pbe alongγ0. Then by total convexity of the set C, ˜γalso lies entirely in C. Ifγ0makes an angle diﬀerent from π atp0, the ﬁrst variation formula will imply that ˜ γis shorter than γ0. This contradicts with the choice of the geodesic loop γ0being the shortest. Now letL: [0,+∞)→Mbe a ray emanating from p0. Chooser >0 large enough and set q=L(r). Consider the distance between qand the geodesic loop γ0. It is clear that the distance can be realized by a geodesic βconnecting the point qto a pointponγ0. LetXbe the unit tangent vector of the geodesic loop γ0atp. ClearlyXis orthogonal to the tangent vector of βatp. We then translate the vector Xalong the geodesic βto get a parallel vector ﬁeld X(t),0≤t≤r. By using this vector ﬁeld we can form a variation ﬁxing one endpoint qand the other on γ0such that the variational vector ﬁeld is (1 −t r)X(t). The second variation of the arclength of this family of curves is given by I/parenleftbigg/parenleftbigg 1−t r/parenrightbigg X(t),/parenleftbigg 1−t r/parenrightbigg X(t)/parenrightbigg =/integraldisplayr 0/bracketleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇∂ ∂t/parenleftbigg/parenleftbigg 1−t r/parenrightbigg X(t)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 −R/parenleftbigg∂ ∂t,/parenleftbigg 1−t r/parenrightbigg X(t),∂ ∂t,/parenleftbigg 1−t r/parenrightbigg X(t)/parenrightbigg/bracketrightbigg dt =1 r−/integraldisplayr 0/parenleftbigg 1−t r/parenrightbigg2 R/parenleftbigg∂ ∂t,X(t),∂ ∂t,X(t)/parenrightbigg dt <0 whenris suﬃciently large, since the sectional curvature of Mis strictly positive everywhere. This contradicts with the fact that βis the shortest geodesic connecting the pointqto the shortest geodesic loop γ0. Thus we have proved the injectivity radius estimate. In contrast to the above injectivity radius estimates, the f ollowing well-known injectivity radius estimate of Cheeger (see for example, Th eorem 5.8 in Cheeger & Ebin [22]) does not impose the restriction on the sign of the s ectional curvature. Cheeger’s Lemma. LetMbe ann-dimensional compact Riemannian manifold with the sectional curvature |KM| ≤λ, the diameter d(M)≤D, and the volume Vol (M)≥v>0. Then, we have inj(M)≥Cn(λ,D,v ) 288 H.-D. CAO AND X.-P. ZHU for some positive constant Cn(λ,D,v )depending only on λ,D,v and the dimension n. For general complete manifolds, it is possible to relate a lo wer injectivity radius bound to some lower volume bound provided one localizes the r elevant geometric quan- tities appropriately. The following injectivity radius es timate, which was ﬁrst obtained by Cheng-Li-Yau [35] for heat kernel estimates and later by C heeger-Gromov-Taylor [27] with a wave equation argument, is a localized version of the above Cheeger’s Lemma. We now present an argument adapted from Abresch and Me yer [1]. Theorem 4.2.2 ( Cheng-Li-Yau [35] ).LetB(x0,4r0),0<r0<∞, be a geodesic ball in ann-dimensional complete Riemannian manifold (M,g)such that the sectional curvatureKof the metric gonB(x0,4r0)satisﬁes the bounds λ≤K≤Λ for some constants λandΛ. Then for any positive constant r≤r0(we will also requirer≤π/(4√ Λ)ifΛ>0) the injectivity radius of Matx0can be bounded from below by inj(M,x 0)≥r·Vol (B(x0,r)) Vol(B(x0,r)) +Vn λ(2r), whereVn λ(2r)denotes the volume of a geodesic ball of radius 2rin then-dimensional simply connected space form Mλwith constant sectional curvature λ. Proof. It is well known (cf. Lemma 5.6 in Cheeger and Ebin [22]) that inj(M,x 0) = min/braceleftbigg conjugate radius of x0,1 2lM(x0)/bracerightbigg wherelM(x0) denotes the length of the shortest (nontrivial) closed geo desic starting and ending at x0. Since by assumption r≤π/(4√ Λ) if Λ>0, the conjugate radius ofx0is at least 4 r. Thus it suﬃces to show (4.2.1) lM(x0)≥2r·Vol (B(x0,r)) Vol (B(x0,r)) +Vn λ(2r). Now we follow the argument presented in [1]. The idea for prov ing this inequality, as indicated in [1], is to compare the geometry of the ball B(x0,4r)⊆B(x0,4r0)⊂ Mwith the geometry of its lifting ˜B4r⊂Tx0(M), via the exponential map expx0, equipped with the pull-back metric ˜ g= exp∗ x0g. Thus expx0:˜B4r→B(x0,4r) is a length-preserving local diﬀeomorphism. Let ˜x0,˜x1,...,˜xNbe the preimages of x0in˜Br⊂˜B4rwith ˜x0= 0. Clearly they one-to-one correspond to the geodesic loops γ0,γ1,...,γ Natx0of length less than r, whereγ0is the trivial loop. Now for each point ˜ xithere exists exactly one isometric immersionϕi:˜Br→˜B4rmapping 0 to ˜ xiand such that expx0ϕi= expx0. Without loss of generality, we may assume γ1is the shortest nontrivial geodesic loop atx0. By analyzing short homotopies, one ﬁnds that ϕi(˜x)/\e}atio\slash=ϕj(˜x) for all ˜x∈˜Br and 0 ≤i<j≤N. This fact has two consequences: (a)N≥2m, wherem= [r/lM(x0)]. To see this, we ﬁrst observe that the points ϕk 1(0),−m≤k≤m, are preimages of x0in˜Brbecauseϕ1is an isometric immersion THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 289 satisfying expx0ϕ1= expx0. Moreover we claim they are distinct. For otherwise ϕ1 would act as a permutation on the set {ϕk 1(0)| −m≤k≤m}. Since the induced metric ˜gat each point in ˜Brhas the injectivity radius at least 2 r, it follows from the Whitehead theorem (see for example [22]) that ˜Bris geodesically convex. Then there would exist the unique center of mass ˜ y∈˜Br. But then ˜ y=ϕ0(˜y) =ϕ1(˜y), a contradiction. (b) Each point in B(x0,r) has at least N+1 preimages in Ω= ∪N i=0B(˜xi,r)⊂˜B2r. Hence by the Bishop volume comparison, (N+ 1)Vol(B(x0,r))≤Vol˜g(Ω)≤Vol˜g(˜B2r)≤Vn λ(2r). Now the inequality (4.2.1) follows by combining the fact N≥2[r/lM(x0)] with the above volume estimate. For our purpose of application, we now consider in a complete Riemannian man- ifoldMa geodesic ball B(p0,s0) (0< s0≤ ∞) with the property that there exists a positive increasing function Λ : [0 ,s0)→[0,∞) such that for any 0 < s < s 0the sectional curvature Kon the ball B(p0,s) of radiussaroundp0satisﬁes the bound |K| ≤Λ(s). Using Theorem 4.2.2, we can control the injectivity radius a t any point p∈B(p0,s0) in terms a positive constant that depends only on the dimensi onn, the injectivity radius at the base point p0, the function Λ and the distance d(p0,p) fromptop0. We now proceed to derive such an estimate. The geometric insi ght of the following argument belongs to Yau [128] where he obtained a lower bound estimate for volume by comparing various geodesic balls. Indeed, it is a ﬁnite ve rsion of Yau’s Busemann function argument which gives the information on comparing geodesic balls with cen- ters far apart. For any point p∈B(p0,s0) withd(p0,p) =s, setr0= (s0−s)/4 (we deﬁne r0=1 ifs0=∞). Deﬁne the set Sto be the union of minimal geodesic segments that connectpto each point in B(p0,r0). Now any point q∈Shas distance at most r0+r0+s=s+ 2r0 fromp0and henceS⊆B(p0,s+2r0). For any 0 <r≤min{π/4/radicalbig Λ(s+ 2r0),r0}, we denote byα(p,r) the sector S∩B(p,r) of radiusrand byα(p,s+r0) =S∩B(p,s+r0). Letα−Λ(s+2r0)(r0) (resp.α−Λ(s+2r0)(s+r0)) be a corresponding sector of the same “angles” with radius r0(resp.s+r0) in then-dimensional simply connected space form with constant sectional curvature −Λ(s+2r0). SinceB(p0,r0)⊂S⊂α(p,s+r0) andα(p,r)⊂B(p,r), the Bishop-Gromov volume comparison theorem implies tha t Vol (B(p0,r0)) Vol(B(p,r))≤Vol(α(p,s+r0)) Vol (α(p,r)) ≤Vol(α−Λ(s+2r0)(s+r0)) Vol (α−Λ(s+2r0)(r))=Vn −Λ(s+2r0)(s+r0) Vn −Λ(s+2r0)(r). Combining this inequality with the local injectivity radiu s estimate in Theorem 4.2.2, we get inj (M,p) ≥rVn −Λ(s+2r0)(r)·Vol (B(p0,r0)) Vn −Λ(s+2r0)(r)Vol (B(p0,r0)) +Vn −Λ(s+2r0)(2r)Vn −Λ(s+2r0)(s+ 2r0). 290 H.-D. CAO AND X.-P. ZHU Thus, we have proved the following Corollary 4.2.3. SupposeB(p0,s0) (0< s0≤ ∞)is a geodesic ball in an n-dimensional complete Riemannian manifold Mhaving the property that for any 0<s<s 0the sectional curvature KonB(p0,s)satisﬁes the bound |K| ≤Λ(s) for some positive increasing function Λdeﬁned on [0,s0). Then for any pointp∈B(p0, s0)withd(p0, p) =sand any positive number r≤ min{π/4/radicalbig Λ(s+ 2r0),r0}withr0= (s0−s)/4, the injectivity radius of Matpis bounded below by inj (M,p) ≥rVn −Λ(s+2r0)(r)·Vol (B(p0,r0)) Vn −Λ(s+2r0)(r)Vol (B(p0,r0)) +Vn −Λ(s+2r0)(2r)Vn −Λ(s+2r0)(s+ 2r0). In particular, we have (4.2.2) inj ( M,p)≥ρn,δ,Λ(s) whereδ >0is a lower bound of the injectivity radius inj(M,p0)at the origin p0 andρn,δ,Λ: [0,s0)→R+is a positive decreasing function that depends only on the dimensionn, the lower bound δof the injectivity radius inj(M,p0), and the function Λ. We remark that in the above discussion if s0=∞then we can apply the standard Bishop relative volume comparison theorem to geodesic ball s directly. Indeed, for any p∈Mand any positive constants randr0, we haveB(p0,r0)⊆B(p,ˆr) with ˆr∆= max{r,r0+d(p0,p)}. Suppose in addition the curvature KonMis uniformly bounded byλ≤K≤Λ for some constants λand Λ, then the Bishop volume comparison theorem implies that Vol(B(p0,r0)) Vol(B(p,r))≤Vol (B(p,ˆr)) Vol (B(p,r))≤Vλ(ˆr) Vλ(r). Hence (4.2.3) inj( M,p)≥rVn λ(r)·Vol (B(p0,r0)) Vn λ(r)Vol (B(p0,r0)) +Vn λ(2r)Vn λ(ˆr). So we see that the injectivity radius inj(M,p) atpfalls oﬀ at worst exponentially as the distance d(p0,p) goes to inﬁnity. In other words, (4.2.4) inj ( M,p)≥c√ B(δ√ B)ne−C√ Bd(p,p0) whereBis an upper bound on the absolute value of the sectional curva ture,δis a lower bound on the injectivity radius at p0withδ <c/√ B, andc>0 andC <+∞ are positive constants depending only on the dimension n. Finally, by combining Theorem 4.2.2 with Perelman’s no loca l collapsing Theo- rem I′(Theorem 3.3.3) we immediately obtain the following import antLittle Loop Lemma conjectured by Hamilton [63]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 291 Theorem 4.2.4 ( Little Loop Lemma ).Letgij(t),0≤t<T < +∞, be a solution of the Ricci ﬂow on a compact manifold M. Then there exists a constant ρ>0having the following property: if at a point x0∈Mand a time t0∈[0,T), |Rm|(·,t0)≤r−2onBt0(x0,r) for somer≤√ T, then the injectivity radius of Mwith respect to the metric gij(t0) atx0is bounded from below by inj (M,x 0,gij(t0))≥ρr. 4.3. Limiting Singularity Models. Consider a solution gij(x,t) of the Ricci ﬂow onM×[0,T),T≤+∞, where either Mis compact or at each time tthe metric gij(·,t) is complete and has bounded curvature. We say that gij(x,t) is amaximal solution if eitherT= +∞orT <+∞and|Rm|is unbounded as t→T. Denote by Kmax(t) = sup x∈M|Rm(x,t)|gij(t). Definition 4.3.1. We say that {(xk,tk)∈M×[0,T)},k= 1,2,..., is a sequence of(almost) maximum points if there exist positive constants c1andα∈(0,1] such that |Rm(xk,tk)| ≥c1Kmax(t), t∈[tk−α Kmax(tk),tk] for allk. Definition 4.3.2. We say that the solution satisﬁes injectivity radius condi- tionif for any sequence of (almost) maximum points {(xk,tk)}, there exists a constant c2>0 independent of ksuch that inj(M,x k,gij(tk))≥c2/radicalbig Kmax(tk)for allk. Clearly, by the Little Loop Lemma, a maximal solution on a com pact manifold with the maximal time T <+∞always satisﬁes the injectivity radius condition. Also by the Gromoll-Meyer injectivity radius estimate, a soluti on on a complete noncom- pact manifold with positive sectional curvature also satis ﬁes the injectivity radius condition. According to Hamilton [63], we classify maximal solutions i nto three types; every maximal solution is clearly of one and only one of the followi ng three types: Type I:T <+∞and sup t∈[0,T)(T−t)Kmax(t)<+∞; Type II: (a)T <+∞but sup t∈[0,T)(T−t)Kmax(t) = +∞; (b)T= +∞but sup t∈[0,T)tKmax(t) = +∞; 292 H.-D. CAO AND X.-P. ZHU Type III: (a)T= +∞,sup t∈[0,T)tKmax(t)<+∞,and limsup t→+∞tKmax(t)>0; (b)T= +∞,sup t∈[0,T)tKmax(t)<+∞,and limsup t→+∞tKmax(t) = 0; It seems that Type III (b) is not compatible with the injectiv ity radius condition unless it is a trivial ﬂat solution. Indeed under the Ricci ﬂo w the length of a curve γ connecting two points x0,x1∈Mevolves by d dtLt(γ) =/integraldisplay γ−Ric (˙γ,˙γ)ds ≤C(n)Kmax(t)·Lt(γ) ≤ǫ tLt(γ),astlarge enough, for arbitrarily ﬁxed ǫ>0. Thus when we are considering the Ricci ﬂow on a compact manifold, the diameter of the evolving manifold grows at mos t astǫ. But the curvature of the evolving manifold decays faster than t−1. This says, as choosing ǫ >0 small enough, diam t(M)2· |Rm(·,t)| →0,ast→+∞. Then it is well-known from Cheeger-Gromov [54] that the mani fold is a nilmanifold and the injectivity radius condition can not be satisﬁed as tlarge enough. When we are considering the Ricci ﬂow on a complete noncompact manif old with nonnegative curvature operator or on a complete noncompact K¨ ahler mani fold with nonnegative holomorphic bisectional curvature, Li-Yau-Hamilton ineq ualities imply that tR(x,t) is increasing in time t. Then Type III(b) occurs only when the solution is a trivial ﬂat metric. For each type of solution we deﬁne a corresponding type of lim iting singularity model. Definition 4.3.3. A solution gij(x,t) to the Ricci ﬂow on the manifold M, where either Mis compact or at each time tthe metric gij(·,t) is complete and has bounded curvature, is called a singularity model if it is not ﬂat and of one of the following three types: Type I : The solution exists for t∈(−∞,Ω) for some constant Ω with 0 <Ω<+∞ and |Rm| ≤Ω/(Ω−t) everywhere with equality somewhere at t= 0; THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 293 Type II : The solution exists for t∈(−∞,+∞) and |Rm| ≤1 everywhere with equality somewhere at t= 0; Type III : The solution exists for t∈(−A,+∞) for some constant Awith 0<A< +∞and |Rm| ≤A/(A+t) everywhere with equality somewhere at t= 0. Theorem 4.3.4. For any maximal solution to the Ricci ﬂow which satisﬁes the injectivity radius condition and is of Type I, II(a), (b), orIII(a), there exists a sequence of dilations of the solution along (almost) maximu m points which converges in theC∞ loctopology to a singularity model of the corresponding type. Proof. Type I: We consider a maximal solution gij(x,t) onM×[0,T) withT <+∞ and Ω∆= limsup t→T(T−t)Kmax(t)<+∞. First we note that Ω >0. Indeed by the evolution equation of curvature, d dtKmax(t)≤Const ·K2 max(t). This implies that Kmax(t)·(T−t)≥Const>0, because limsup t→TKmax(t) = +∞. Thus Ω must be positive. Choose a sequence of points xkand timestksuch thattk→Tand lim k→∞(T−tk)|Rm(xk,tk)|= Ω. Denote by ǫk=1/radicalbig |Rm(xk,tk)|. We translate in time so that tkbecomes 0, dilate in space by the factor ǫkand dilate in time byǫ2 kto get ˜g(k) ij(·,˜t) =ǫ−2 kgij(·,tk+ǫ2 k˜t),˜t∈[−tk/ǫ2 k,(T−tk)/ǫ2 k). 294 H.-D. CAO AND X.-P. ZHU Then ∂ ∂˜t˜g(k) ij(·,˜t) =ǫ−2 k∂ ∂tgij(·,t)·ǫ2 k =−2Rij(·,tk+ǫ2 k˜t) =−2˜R(k) ij(·,˜t), where ˜R(k) ijis the Ricci curvature of the metric ˜ g(k) ij. So ˜g(k) ij(·,˜t) is still a solution to the Ricci ﬂow which exists on the time interval [ −tk/ǫ2 k,(T−tk)/ǫ2 k), where tk/ǫ2 k=tk|Rm(xk,tk)| →+∞ and (T−tk)/ǫ2 k= (T−tk)|Rm(xk,tk)| →Ω. For anyǫ>0 we can ﬁnd a time τ <T such that for t∈[τ,T), |Rm| ≤(Ω +ǫ)/(T−t) by the assumption. Then for ˜t∈[(τ−tk)/ǫ2 k,(T−tk)/ǫ2 k), the curvature of ˜ g(k) ij(·,˜t) is bounded by |˜Rm(k)|=ǫ2 k|Rm| ≤(Ω +ǫ)/((T−t)|Rm(xk,tk)|) = (Ω +ǫ)/((T−tk)|Rm(xk,tk)|+ (tk−t)|Rm(xk,tk)|) →(Ω +ǫ)/(Ω−˜t),ask→+∞. This implies that {(xk,tk)}is a sequence of (almost) maximum points. And then by the injectivity radius condition and Hamilton’s compact ness theorem 4.1.5, there exists a subsequence of the metrics ˜ g(k) ij(˜t) which converges in the C∞ loctopology to a limit metric ˜ g(∞) ij(˜t) on a limiting manifold ˜Mwith˜t∈(−∞,Ω) such that ˜ g(∞) ij(˜t) is a complete solution of the Ricci ﬂow and its curvature satisﬁ es the bound |˜Rm(∞)| ≤Ω/(Ω−˜t) everywhere on ˜M×(−∞,Ω) with the equality somewhere at ˜t= 0. Type II(a): We consider a maximal solution gij(x,t) onM×[0,T) with T <+∞and limsup t→T(T−t)Kmax(t) = +∞. LetTk< T < +∞withTk→T, andγkր1, ask→+∞. Pick points xkand timestksuch that, as k→+∞, (Tk−tk)|Rm(xk,tk)| ≥γksup x∈M,t≤Tk(Tk−t)|Rm(x,t)| →+∞. Again denote by ǫk=1/radicalbig |Rm(xk,tk)| THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 295 and dilate the solution as before to get ˜g(k) ij(·,˜t) =ǫ−2 kgij(·,tk+ǫ2 k˜t),˜t∈[−tk/ǫ2 k,(Tk−tk)/ǫ2 k), which is still a solution to the Ricci ﬂow and satisﬁes the cur vature bound |˜Rm(k)|=ǫ2 k|Rm| ≤1 γk·(Tk−tk) (Tk−t) =1 γk(Tk−tk)|Rm(xk,tk)| [(Tk−tk)|Rm(xk,tk)| −˜t]for˜t∈/bracketleftbigg −tk ǫ2 k,(Tk−tk) ǫ2 k/parenrightbigg , sincet=tk+ǫ2 k˜tandǫk= 1//radicalbig |Rm(xk,tk)|. Hence {(xk,tk)}is a sequence of (almost) maximum points. And then as before, by applying Hamilton’s c ompactness theorem 4.1.5, there exists a subsequence of the metrics ˜ g(k) ij(˜t) which converges in the C∞ loc topology to a limit ˜ g(∞) ij(˜t) on a limiting manifold ˜Mand˜t∈(−∞,+∞) such that ˜g(∞) ij(˜t) is a complete solution of the Ricci ﬂow and its curvature sat isﬁes |˜Rm(∞)| ≤1 everywhere on ˜M×(−∞,+∞) and the equality holds somewhere at ˜t= 0. Type II(b): We consider a maximal solution gij(x,t) onM×[0,T) with T= +∞and limsup t→TtKmax(t) = +∞. Again letTk→T= +∞, andγkր1, ask→+∞. Pickxkandtksuch that tk(Tk−tk)|Rm(xk,tk)| ≥γksup x∈M,t≤Tkt(Tk−t)|Rm(x,t)|. Deﬁne ˜g(k) ij(·,˜t) =ǫ−2 kgij(·,tk+ǫ2 k˜t),˜t∈[−tk/ǫ2 k,(Tk−tk)/ǫ2 k), whereǫk= 1//radicalbig |Rm(xk,tk)|. Since tk(Tk−tk)|Rm(xk,tk)| ≥γksup x∈M,t≤Tkt(Tk−t)|Rm(x,t)| ≥γksup x∈M,t≤Tk/2t(Tk−t)|Rm(x,t)| ≥Tk 2γksup x∈M,t≤Tk/2t|Rm(x,t)|, we have tk ǫ2 k=tk|Rm(xk,tk)| ≥γk 2/parenleftbiggTk Tk−tk/parenrightbigg sup x∈M,t≤Tk/2t|Rm(x,t)| →+∞, and (Tk−tk) ǫ2 k= (Tk−tk)|Rm(xk,tk)| ≥γk 2/parenleftbiggTk tk/parenrightbigg sup x∈M,t≤Tk/2t|Rm(x,t)| →+∞, 296 H.-D. CAO AND X.-P. ZHU ask→+∞. As before, we also have ∂ ∂˜t˜g(k) ij(·,˜t) =−2˜R(k) ij(·,˜t) and |˜Rm(k)| =ǫ2 k|Rm| ≤1 γk·tk(Tk−tk) t(Tk−t) =1 γk·tk(Tk−tk)|Rm(xk, tk)| (tk+ǫ2 k˜t)[(Tk−tk)−ǫ2 k˜t]· |Rm(xk, tk)| =1 γk·tk(Tk−tk)|Rm(xk, tk)| (tk+ǫ2 k˜t)[(Tk−tk)|Rm(xk, tk)| −˜t] =tk(Tk−tk)|Rm(xk, tk)| γk( 1+˜t/(tk|Rm(xk, tk)|))[tk(Tk−tk)|Rm(xk, tk)|](1−˜t/((Tk−tk)|Rm(xk, tk)|)) →1,ask→+∞. Hence {(xk,tk)}is again a sequence of (almost) maximum points. As before, th ere exists a subsequence of the metrics ˜ g(k) ij(˜t) which converges in the C∞ loctopology to a limit ˜g(∞) ij(˜t) on a limiting manifold ˜Mand˜t∈(−∞,+∞) such that ˜ g(∞) ij(˜t) is a complete solution of the Ricci ﬂow and its curvature satisﬁe s |˜Rm(∞)| ≤1 everywhere on ˜M×(−∞,+∞) with the equality somewhere at ˜t= 0. Type III(a): We consider a maximal solution gij(x,t) onM×[0,T) with T= +∞and limsup t→TtKmax(t) =A∈(0,+∞). Choose a sequence of xkandtksuch thattk→+∞and lim k→∞tk|Rm(xk,tk)|=A. Setǫk= 1//radicalbig |Rm(xk,tk)|and dilate the solution as before to get ˜g(k) ij(·,˜t) =ǫ−2 kgij(·,tk+ǫ2 k˜t),˜t∈[−tk/ǫ2 k,+∞) which is still a solution to the Ricci ﬂow. Also, for arbitrar ily ﬁxedǫ>0, there exists a suﬃciently large positive constant τsuch that for t∈[τ,+∞), |˜Rm(k)|=ǫ2 k|Rm| ≤ǫ2 k/parenleftbiggA+ǫ t/parenrightbigg =ǫ2 k/parenleftbiggA+ǫ tk+ǫ2 k˜t/parenrightbigg = (A+ǫ)/(tk|Rm(xk,tk)|+˜t),for˜t∈[(τ−tk)/ǫ2 k,+∞). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 297 Note that (A+ǫ)/(tk|Rm(xk,tk)|+˜t)→(A+ǫ)/(A+˜t),ask→+∞ and (τ−tk)/ǫ2 k→ −A,ask→+∞. Hence {(xk,tk)}is a sequence of (almost) maximum points. And then as before, there exists a subsequence of the metrics ˜ g(k) ij(˜t) which converges in the C∞ loctopology to a limit ˜g(∞) ij(˜t) on a limiting manifold ˜Mand˜t∈(−A,+∞) such that ˜ g(∞) ij(˜t) is a complete solution of the Ricci ﬂow and its curvature satisﬁe s |˜Rm(∞)| ≤A/(A+˜t) everywhere on ˜M×(−A,+∞) with the equality somewhere at ˜t= 0. In the case of manifolds with nonnegative curvature operato r, or K¨ ahler metrics with nonnegative holomorphic bisectional curvature, we ca n bound the Riemannian curvature by the scalar curvature Rupto a constant factor depending only on the dimension. Then we can slightly modify the statements in the previous theorem as follows Corollary 4.3.5. For any complete maximal solution to the Ricci ﬂow with bounded and nonnegative curvature operator on a Riemannian manifold, or on a K¨ ahler manifold with bounded and nonnegative holomorphic bisectional curvature, there exists a sequence of dilations of the solution along (almost )maximum points which converges to a singular model. For Type Isolutions: the limit model exists for t∈(−∞,Ω)with0<Ω<+∞and has R≤Ω/(Ω−t) everywhere with equality somewhere at t= 0. For Type IIsolutions: the limit model exists for t∈(−∞,+∞)and has R≤1 everywhere with equality somewhere at t= 0. For Type IIIsolutions: the limit model exists for t∈(−A,+∞)with0< A < +∞ and has R≤A/(A+t) everywhere with equality somewhere at t= 0. A natural and important question is to understand each of the three types of singularity models. The following results obtained by Hami lton [66] and Chen-Zhu [30] characterize the Type II and Type III singularity models wit h nonnegative curvature operator and positive Ricci curvature respectively. The co rresponding results in the K¨ ahler case with nonnegative holomorphic bisectional cur vature were obtained by the ﬁrst author [14]. 298 H.-D. CAO AND X.-P. ZHU Theorem 4.3.6. (i)(Hamilton [66]) Any Type IIsingularity model with nonnegative curvature operator and positive Ricci curvature to the Ricci ﬂow on a ma nifoldMmust be a(steady )Ricci soliton. (ii)(Chen-Zhu [30]) Any Type IIIsingularity model with nonnegative curvature operator and positive Ricci curvature on a manifold Mmust be a homotheti- cally expanding Ricci soliton. Proof. We only give the proof of (ii), since the proof of (i) is simila r and easier. After a shift of the time variable, we may assume the Type III s ingularity model is deﬁned on 0 <t< +∞andtRassumes its maximum in space-time. Recall from the Li-Yau-Hamilton inequality (Theorem 2.5.4 ) that for any vectors ViandWi, (4.3.1) MijWiWj+ (Pkij+Pkji)VkWiWj+RikjlWiWjVkVl≥0, where Mij= ∆Rij−1 2∇i∇jR+ 2RipjqRpq−gpqRipRjq+1 2tRij and Pijk=∇iRjk− ∇ jRik. Take the trace on Wto get (4.3.2) Q∆=∂R ∂t+R t+ 2∇iR·Vi+ 2RijViVj≥0 for any vector Vi. Let us choose Vto be the vector ﬁeld minimizing Q, i.e., (4.3.3) Vi=−1 2(Ric−1)ik∇kR, where (Ric−1)ikis the inverse of the Ricci tensor Rij. Substitute this vector ﬁeld VintoQto get a smooth function ˜Q. By a direct computation from the evolution equations of curvatures (see [61] for details), (4.3.4)∂ ∂t˜Q≥∆˜Q−2 t˜Q. SupposetRassumes its maximum at ( x0,t0) witht0>0, then ∂R ∂t+R t= 0,at (x0,t0). This implies that the quantity Q=∂R ∂t+R t+ 2∇iR·Vi+ 2RijViVj vanishes in the direction V= 0 at (x0,t0). We claim that for any earlier time t<t 0 and any point x∈M, there is a vector V∈TxMsuch thatQ= 0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 299 We argue by contradiction. Suppose not, then there is ¯ x∈Mand 0<¯t<t 0such that˜Qis positive at x= ¯xandt=¯t. We can ﬁnd a nonnegative smooth function ρ onMwith support in a neighborhood of ¯ xso thatρ(¯x)>0 and ˜Q≥ρ ¯t2, att=¯t. Letρevolve by the heat equation ∂ρ ∂t= ∆ρ. It then follows from the standard strong maximum principle t hatρ >0 everywhere for anyt>¯t. From (4.3.4) we see that ∂ ∂t/parenleftig ˜Q−ρ t2/parenrightig ≥∆/parenleftig ˜Q−ρ t2/parenrightig −2 t/parenleftig ˜Q−ρ t2/parenrightig Then by the maximum principle as in Chapter 2, we get ˜Q≥ρ t2>0,for allt≥¯t. This gives a contradiction with the fact Q= 0 forV= 0 at (x0,t0). We thus prove the claim. Consider each time t<t 0. The null vector ﬁeld of Qsatisﬁes the equation (4.3.5) ∇iR+ 2RijVj= 0, by the ﬁrst variation of QinV. SinceRijis positive, we see that such a null vector ﬁeld is unique and varies smoothly in space-time. Substituting (4.3.5) into the expression of Q, we have (4.3.6)∂R ∂t+R t+∇iR·Vi= 0 Denote by Qij=Mij+ (Pkij+Pkji)Vk+RikjlVkVl. From (4.3.1) we see that Qijis nonnegative deﬁnite with its trace Q= 0 for such a null vector V. It follows that Qij=Mij+ (Pkij+Pkji)Vk+RikjlVkVl= 0. Again from the ﬁrst variation of QijinV, we see that (4.3.7) ( Pkij+Pkji) + (Rikjl+Rjkil)Vl= 0, and hence (4.3.8) Mij−RikjlVkVl= 0. Applying the heat operator to (4.3.5) and (4.3.6) we get 0 =/parenleftbigg∂ ∂t−∆/parenrightbigg (∇iR+ 2RijVj) (4.3.9) = 2Rij/parenleftbigg∂ ∂t−∆/parenrightbigg Vj+/parenleftbigg∂ ∂t−∆/parenrightbigg (∇iR) + 2Vj/parenleftbigg∂ ∂t−∆/parenrightbigg Rij−4∇kRij∇kVj, 300 H.-D. CAO AND X.-P. ZHU and 0 =/parenleftbigg∂ ∂t−∆/parenrightbigg/parenleftbigg∂R ∂t+R t+∇iR·Vi/parenrightbigg (4.3.10) =∇iR/parenleftbigg∂ ∂t−∆/parenrightbigg Vi+Vi/parenleftbigg∂ ∂t−∆/parenrightbigg (∇iR)−2∇k∇iR· ∇kVi +/parenleftbigg∂ ∂t−∆/parenrightbigg/parenleftbigg∂R ∂t+R t/parenrightbigg . Multiplying (4.3.9) by Vi, summing over iand adding (4.3.10), as well as using the evolution equations on curvature, we get 0 = 2Vi(2∇i(|Rc|2)−Ril∇lR) + 2ViVj(2RpiqjRpq−2gpqRpiRqj) (4.3.11) −4∇kRij· ∇kVj·Vi−2∇k∇iR· ∇kVi+ 4Rij∇i∇jR + 4gklgmngpqRkmRnpRql+ 4RijklRikVjVl−R t2. From (4.3.5), we have the following equalities (4.3.12)  −2ViRil∇lR−4ViVjgpqRpiRqj= 0, −4∇kRij· ∇kVj·Vi−2∇k∇iR· ∇kVi= 4Rij∇kVi· ∇kVj, ∇i∇jR=−2∇iRjl·Vl−2Rjl∇iVl. Substituting (4.3.12) into (4.3.11), we obtain 8Rij(∇kRij·Vk+RikjlVkVl− ∇ iRjl·Vl−Rjl∇iVl) + 4Rij∇kVi· ∇kVj+ 4gklgmngpqRkmRnpRql−R t2= 0. By using (4.3.7), we know Rij(∇kRij·Vk+RikjlVkVl− ∇ iRjl·Vl) = 0. Then we have (4.3.13) −8RijRjl∇iVl+ 4Rij∇kVi· ∇kVj+ 4gklgmngpqRkmRnpRql−R t2= 0. By taking the trace in the last equality in (4.3.12) and using (4.3.6) and the evolution equation of the scalar curvature, we can get (4.3.14) Rij(Rij+gij 2t− ∇ iVj) = 0. Finally by combining (4.3.13) and (4.3.14), we deduce 4Rijgkl/parenleftig Rik+gik 2t− ∇ kVi/parenrightig/parenleftig Rjk+gjk 2t− ∇ kVj/parenrightig = 0. SinceRijis positive deﬁnite, we get (4.3.15) ∇iVj=Rij+gij 2t,for alli,j. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 301 This means that gij(t) is a homothetically expanding Ricci soliton. Remark 4.3.7. Recall from Section 1.5 that any compact steady Ricci solito n or expanding Ricci soliton must be Einstein. If the manifold Min Theorem 4.3.6 is noncompact and simply connected, then the steady (or expand ing) Ricci soliton must be a steady (or expanding) gradient Ricci soliton. For examp le, we know that ∇iVj is symmetric from (4.3.15). Also, by the simply connectedne ss ofMthere exists a functionFsuch that ∇i∇jF=∇iVj,onM. So Rij=∇i∇jF−gij 2t,onM This means that gijis an expanding gradient Ricci soliton. In the K¨ ahler case, we have the following results for Type II and Type III sin- gularity models with nonnegative holomorphic bisectional curvature obtained by the ﬁrst author in [14]. Theorem 4.3.8 ( Cao [14] ). (i) Any Type IIsingularity model on a K¨ ahler manifold with nonnegative ho lo- morphic bisectional curvature and positive Ricci curvatur e must be a steady K¨ ahler-Ricci soliton. (ii) Any Type IIIsingularity model on a K¨ ahler manifold with nonnegative ho lo- morphic bisectional curvature and positive Ricci curvatur e must be an expand- ing K¨ ahler-Ricci soliton. To conclude this section, we state a result of Sesum [113] on c ompact Type I singularity models. Recall that Perelman’s functional W, introduced in Section 1.5, is given by W(g,f,τ) =/integraldisplay M(4πτ)−n 2[τ(|∇f|2+R) +f−n]e−fdVg with the function fsatisfying the constraint /integraldisplay M(4πτ)−n 2e−fdVg= 1. And recall from Corollary 1.5.9 that µ(g(t)) = inf/braceleftbigg W(g(t),f,T−t)|/integraldisplay M(4π(T−t))−n 2e−fdVg(t)= 1/bracerightbigg is strictly increasing along the Ricci ﬂow unless we are on a g radient shrinking soliton. If one can show that µ(g(t)) is uniformly bounded from above and the minimizing functionsf=f(·,t) have a limit as t→T, then the rescaling limit model will be a shrinking gradient soliton. As shown by Natasa Sesum in [11 3], Type I assump- tion guarantees the boundedness of µ(g(t)), while the compactness assumption of the rescaling limit guarantees the existence of the limit fo r the minimizing functions f(·,t). Therefore we have 302 H.-D. CAO AND X.-P. ZHU Theorem 4.3.9 ( Sesum [113] ).Let(M,g ij(t))be a Type Isingularity model obtained as a rescaling limit of a Type Imaximal solution. Suppose Mis compact. Then (M,g ij(t))must be a gradient shrinking Ricci soliton. It seems that the assumption on the compactness of the rescal ing limit is super- ﬂuous. We conjecture that any noncompact Type I limit is also a gradient shrinking soliton. 4.4. Ricci Solitons. We will now examine the structure of a steady Ricci soliton of the sort we get as a Type II limit. Lemma 4.4.1. Suppose we have a complete gradient steady Ricci soliton gijwith bounded curvature so that Rij=∇i∇jF for some function FonM. Assume the Ricci curvature is positive and the scalar curvatureRattains its maximum Rmaxat a pointx0∈M. Then (4.4.1) |∇F|2+R=Rmax everywhere on M, and furthermore Fis convex and attains its minimum at x0. Proof. Recall that, from (1.1.15) and noting our Fhere is −fthere, the steady gradient Ricci soliton has the property |∇F|2+R=C0 for some constant C0. Clearly,C0≥Rmax. IfC0=Rmax, then ∇F= 0 at the point x0. Since ∇i∇jF=Rij>0,we see thatFis convex and Fattains its minimum at x0. IfC0>Rmax, consider a gradient path of Fin a local coordinate neighborhood throughx0= (x1 0,...,xn 0) : /braceleftigg xi=xi(u), u ∈(−ε,ε), i= 1,...,n xi 0=xi(0), and dxi du=gij∇jF, u ∈(−ε,ε). Now|∇F|2=C0−R≥C0−Rmax>0 everywhere, while |∇F|2is smallest at x=x0 sinceRis largest there. But we compute d du|∇F|2= 2gjl/parenleftbiggd du∇jF/parenrightbigg ∇lF = 2gikgjl∇i∇jF· ∇kF∇lF = 2gikgjlRij∇kF∇lF >0 sinceRij>0 and |∇F|2>0. Then |∇F|2is not smallest at x0, and we have a contradiction. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 303 We remark that when we are considering a complete expanding g radient Ricci soliton onMwith positive Ricci curvature and Rij+ρgij=∇i∇jF for some constant ρ>0 and some function F, the above argument gives |∇F|2+R−2ρF=C for some positive constant C. Moreover the function Fis an exhausting and convex function. In particular, such an expanding gradient Ricci s oliton is diﬀeomorphic to the Euclidean space Rn. Let us introduce a geometric invariant as follows. Let Obe a ﬁxed point in a Riemannian manifold M,sthe distance to the ﬁxed point O, andRthe scalar curvature. We deﬁne the asymptotic scalar curvature ratio A= limsup s→+∞Rs2. Clearly the deﬁnition is independent of the choice of the ﬁxe d pointOand invariant under dilation. This concept is particular useful on manifo lds with positive sectional curvature. The ﬁrst type of gap theorem was obtained by Mok-S iu-Yau [93] in un- derstanding the hypothesis of the paper of Siu-Yau [120]. Ya u (see [49]) suggested that this should be a general phenomenon. This was later conf ormed by Greene-Wu [49, 50], Eschenberg-Shrader-Strake [45] and Drees [44] wh ere they show that any complete noncompact n-dimensional (except n= 4 or 8) Riemannian manifold of positive sectional curvature must have A >0. Similar results on complete noncom- pact K¨ ahler manifolds of positive holomorphic bisectiona l curvature were obtained by Chen-Zhu [31] and Ni-Tam [100]. Theorem 4.4.2 ( Hamilton [63] ).For a complete noncompact steady gradient Ricci soliton with bounded curvature and positive sectiona l curvature of dimension n≥ 3where the scalar curvature assume its maximum at a point O∈M, the asymptotic scalar curvature ratio is inﬁnite, i.e., A= limsup s→+∞Rs2= +∞ where s is the distance to the point O. Proof. The solution to the Ricci ﬂow corresponding to the soliton ex ists for −∞<t< +∞and is obtained by ﬂowing along the gradient of a potential fu nction Fof the soliton. We argue by contradiction. Suppose Rs2≤C. We will show that the limit ¯gij(x) = lim t→−∞gij(x,t) exists forx/\e}atio\slash=Oon the manifold Mand is a complete ﬂat metric on M\ {O}.Since the sectional curvature of Mis positive everywhere, it follows from Cheeger-Gromoll [23] thatMis diﬀeomorphic to Rn. ThusM\{O}is diﬀeomorphic to Sn−1×R. But forn≥3 there is no ﬂat metric on Sn−1×R, and this will ﬁnish the proof. 304 H.-D. CAO AND X.-P. ZHU To see the limit metric exists, we note that R→0 ass→+∞, so|∇F|2→Rmax ass→+∞by (4.4.1). The function Fitself can be taken to evolve with time, using the deﬁnition ∂F ∂t=∇iF·∂xi ∂t=−|∇F|2= ∆F−Rmax which pulls Fback by the ﬂow along the gradient of F. Then we continue to have ∇i∇jF=Rijfor all time, and |∇F|2→Rmaxass→+∞for each time. When we go backward in time, this is equivalent to ﬂowing outw ards along the gradient of F, and our speed approaches√Rmax. So, starting outside of any neigh- borhood of Owe have s |t|=dt(·,O) |t|→/radicalbig Rmax,ast→ −∞ and (4.4.2) R(·,t)≤C Rmax· |t|2,as|t|large enough . Hence for |t|suﬃciently large, 0≥ −2Rij =∂ ∂tgij ≥ −2Rgij ≥ −2C Rmax· |t|2gij which implies that for any tangent vector V, 0≤d d|t|(log(gij(t)ViVj))≤2C Rmax· |t|2. These two inequalities show that gij(t)ViVjhas a limit ¯ gijViVjast→ −∞ . Since the metrics are all essentially the same, it always tak es an inﬁnite length to get out to the inﬁnity. This shows the limit ¯ gijis complete at the inﬁnity. One the other hand, any point Pother than Owill eventually be arbitrarily far from O, so the limit metric ¯ gijis also complete away from OinM\ {O}. Using Shi’s derivative estimates in Chapter 1, it follows that gij(·,t) converges in the C∞ loctopology to a complete smooth limit metric ¯ gijast→ −∞ , and the limit metric is ﬂat by (4.4.2). The above argument actually shows that (4.4.3) limsup s→+∞Rs1+ε= +∞ for arbitrarily small ε>0 and for any complete gradient Ricci soliton with bounded and positive sectional curvature of dimension n≥3 where the scalar curvature as- sumes its maximum at a ﬁxed point O. Finally we conclude this section with the important uniquen ess of complete Ricci soliton on two-dimensional Riemannian manifolds. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 305 Theorem 4.4.3 ( Hamilton [60] ).The only complete steady Ricci soliton on a two-dimensional manifold with bounded curvature which ass umes its maximum 1at an origin is the “cigar” soliton on the plane R2with the metric ds2=dx2+dy2 1 +x2+y2. Proof. Recall that the scalar curvature evolves by ∂R ∂t= ∆R+R2 on a two-dimensional manifold M. Denote by Rmin(t) = inf {R(x,t)|x∈M}.We see from the maximum principle (see for example Chapter 2) th atRmin(t) is strictly increasing whenever Rmin(t)/\e}atio\slash= 0, for −∞<t< +∞. This shows that the curvature of a steady Ricci soliton on a two-dimensional manifold Mmust be nonnegative and Rmin(t) = 0 for all t∈(−∞,+∞). Further by the strong maximum principle we see that the curvature is actually positive everywhere. In part icular, the manifold must be noncompact. So the manifold Mis diﬃomorphic to R2and the Ricci soliton must be a gradient soliton. Let Fbe a potential function of the gradient Ricci soliton. Then, by deﬁnition, we have ∇iVj+∇jVi=Rgij withVi=∇iF. This says that the vector ﬁeld Vmust be conformal. In complex coordinate a conformal vector ﬁeld is holomorphic. Hence Vis locally given by V(z)∂ ∂z for a holomorphic function V(z). At a zero of Vthere will be a power series expansion V(z) =azp+···,(a/\e}atio\slash= 0) and ifp>1 the vector ﬁeld will have closed orbits in any neighborhood of the zero. Now the vector ﬁeld is gradient and a gradient ﬂow cannot have a closed orbit. Hence V(z) has only simple zeros. By Lemma 4.4.1, we know that Fis strictly convex with the only critical point being the minima, chosen to be th e origin of R2. So the holomorphic vector ﬁeld Vmust be V(z)∂ ∂z=cz∂ ∂z,forz∈C, for some complex number c. We now claim that cis real. Let us write the metric as ds2=g(x,y)(dx2+dy2) withz=x+√−1y.Then∇F=cz∂ ∂zmeans that if c=a+√−1b, then ∂F ∂x= (ax−by)g,∂F ∂y= (bx+ay)g. Taking the mixed partial derivatives∂2F ∂x∂yand equating them at the origin x=y= 0 givesb= 0, socis real. 306 H.-D. CAO AND X.-P. ZHU Let /braceleftigg x=eucosv, −∞<u< +∞, y=eusinv, 0≤v≤2π. Write ds2=g(x,y)(dx2+dy2) =g(eucosv,eusinv)e2u(du2+dv2) ∆=g(u,v)(du2+dv2). Then we get the equations ∂F ∂u=ag,∂F ∂v= 0 since the gradient of Fis justa∂ ∂ufor a real constant a. The second equation shows thatF=F(u) is a function of uonly, then the ﬁrst equation shows that g=g(u) is also a function of uonly. Then we can write the metric as ds2=g(u)(du2+dv2) (4.4.4) =g(u)e−2u(dx2+dy2). This implies that e−2ug(u) must be a smooth function of x2+y2=e2u. So as u→ −∞ , (4.4.5) g(u) =b1e2u+b2(e2u)2+···, withb1>0. The curvature of the metric is given by R=−1 g/parenleftbiggg′ g/parenrightbigg′ where ( ·)′is the derivative with respect to u. Note that the soliton is by translation inuwith velocity c. Henceg=g(u+ct) satisﬁes ∂g ∂t=−Rg which becomes cg′=/parenleftbiggg′ g/parenrightbigg′ . Thus by (4.4.5), g′ g=cg+ 2 and then by integrating e2u/parenleftbigg1 g/parenrightbigg =−c 2e2u+b1 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 307 i.e., g(u) =e2u b1−c 2e2u. In particular, we have c<0 since the Ricci soliton is not ﬂat. Therefore ds2=g(u)e−2u(dx2+dy2) =dx2+dy2 α1+α2(x2+y2) for some constants α1,α2>0. By the normalization condition that the curvature attains its maximum 1 at the origin, we conclude that ds2=dx2+dy2 1 + (x2+y2). 5. Long Time Behaviors. LetMbe a complete manifold of dimension n. Consider a solution of the Ricci ﬂow gij(x,t) onMand on a maximal time interval [0,T). WhenMis compact, we usually consider the normalized Ricci ﬂow ∂gij ∂t=2 nrgij−2Rij, wherer=/integraltext MRdV//integraltext MdVis the average scalar curvature. The factor rserves to normalize the Ricci ﬂow so that the volume is constant. To see this we observe that dV=/radicalbig detgijdxand then ∂ ∂tlog/radicalbig detgij=1 2gij∂ ∂tgij=r−R, d dt/integraldisplay MdV=/integraldisplay M(r−R)dV= 0. The Ricci ﬂow and the normalized Ricci ﬂow diﬀer only by a chan ge of scale in space and a change of parametrization in time. Indeed, we ﬁrst assu me thatgij(t) evolves by the (unnormalized) Ricci ﬂow and choose the normalizatio n factorψ=ψ(t) so that ˜gij=ψgij, and/integraltext Md˜µ= 1. Next we choose a new time scale ˜t=/integraltext ψ(t)dt. Then for the normalized metric ˜ gijwe have ˜Rij=Rij,˜R=1 ψR,˜r=1 ψr. Because/integraltext Md˜V= 1, we see that/integraltext MdV=ψ−n 2. Then d dtlogψ=/parenleftbigg −2 n/parenrightbiggd dtlog/integraldisplay MdV =/parenleftbigg −2 n/parenrightbigg/integraltext M∂ ∂t/radicalbig detgijdx/integraltext MdV =2 nr, 308 H.-D. CAO AND X.-P. ZHU since∂ ∂tgij=−2Rijfor the Ricci ﬂow. Hence it follows that ∂ ∂˜t˜gij=∂ ∂tgij+/parenleftbiggd dtlogψ/parenrightbigg gij =2 n˜r˜gij−2˜Rij. Thus studying the behavior of the Ricci ﬂow near the maximal t ime is equivalent to studying the long-time behavior of the normalized Ricci ﬂow . In this chapter we will obtain long-time behavior of the norm alized Ricci ﬂow for the following special cases: (1) compact two-manifolds; (2 ) compact three-manifolds with nonnegative Ricci curvature; (3) compact four-manifo lds with nonnegative cur- vature operator; and (4) compact three-manifolds with unif ormly bounded normalized curvature. 5.1. The Ricci Flow on Two-manifolds. LetMbe a compact surface, we will discuss in this section the evolution of a Riemannian metric gijunder the normalized Ricci ﬂow. On a surface, the Ricci curvature is given by Rij=1 2Rgij so the normalized Ricci ﬂow equation becomes (5.1.1)∂ ∂tgij= (r−R)gij. Recall the Gauss-Bonnet formula says /integraldisplay MRdV= 4πχ(M), whereχ(M) is the Euler characteristic number of M. Thus the average scalar curva- turer= 4πχ(M)//integraltext MdVis constant in time. To obtain the evolution equation of the normalized curvatur e, we recall a simple principle in [58] for converting from the unnormalized to th e normalized evolution equation on an n-dimensional manifold. Let PandQbe two expressions formed from the metric and curvature tensors, and let ˜Pand˜Qbe the corresponding expressions for the normalized Ricci ﬂow. Since they diﬀer by dilations, they diﬀer by a power of the normalized factor ψ=ψ(t). We sayPhasdegreekif˜P=ψkP. Thusgijhas degree 1,Rijhas degree 0, Rhas degree −1. Lemma 5.1.1. SupposePsatisﬁes ∂P ∂t= ∆P+Q for the unnormalized Ricci ﬂow, and Phas degreek. ThenQhas degreek−1, and for the normalized Ricci ﬂow, ∂˜P ∂˜t=˜∆˜P+˜Q+2 nk˜r˜P. Proof. We ﬁrst see Qhas degree k−1 since∂˜t/∂t=ψand ∆ =ψ˜∆. Then ψ∂ ∂˜t(ψ−k˜P) =ψ˜∆(ψ−k˜P) +ψ−k+1˜Q THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 309 which implies ∂˜P ∂˜t=˜∆˜P+˜Q+k ψ∂ψ ∂˜t˜P =˜∆˜P+˜Q+2 nk˜r˜P since∂ ∂˜tlogψ= (∂ ∂tlogψ)ψ−1=2 n˜r. We now come back to the normalized Ricci ﬂow (5.1.1) on a compa ct surface. By applying the above lemma to the evolution equation of unnorm alized scalar curvature, we have (5.1.2)∂R ∂t= ∆R+R2−rR for the normalized scalar curvature R. As a direct consequence, by using the maximum principle, both nonnegative scalar curvature and nonposit ive scalar curvature are preserved for the normalized Ricci ﬂow on surfaces. Let us introduce a potential function ϕas in the K¨ ahler-Ricci ﬂow (see for example [11]). Since R−rhas mean value zero on a compact surface, there exists a uniqu e functionϕ, with mean value zero, such that (5.1.3) ∆ ϕ=R−r. Diﬀerentiating (5.1.3) in time, we have ∂ ∂tR=∂ ∂t(∆ϕ) = (R−r)∆ϕ+gij∂ ∂t/parenleftbigg∂2ϕ ∂xi∂xj−Γk ij∂ϕ ∂xk/parenrightbigg = (R−r)∆ϕ+ ∆/parenleftbigg∂ϕ ∂t/parenrightbigg . Combining with the equation (5.1.2), we get ∆/parenleftbigg∂ϕ ∂t/parenrightbigg = ∆(∆ϕ) +r∆ϕ which implies that (5.1.4)∂ϕ ∂t= ∆ϕ+rϕ−b(t) for some function b(t) of time only. Since/integraltext MϕdV= 0 for allt, we have 0 =d dt/integraldisplay Mϕdµ=/integraldisplay M(∆ϕ+rϕ−b(t))dµ+/integraldisplay Mϕ(r−R)dµ =−b(t)/integraldisplay Mdµ+/integraldisplay M|∇ϕ|2dµ. Thus the function b(t) is given by b(t) =/integraltext M|∇ϕ|2dµ/integraltext Mdµ. 310 H.-D. CAO AND X.-P. ZHU Deﬁne a function hby h= ∆ϕ+|∇ϕ|2= (R−r) +|∇ϕ|2, and set Mij=∇i∇jϕ−1 2∆ϕgij to be the traceless part of ∇i∇jϕ. Lemma 5.1.2. The function hsatisﬁes the evolution equation (5.1.5)∂h ∂t= ∆h−2|Mij|2+rh. Proof. Under the normalized Ricci ﬂow, ∂ ∂t|∇ϕ|2=/parenleftbigg∂ ∂tgij/parenrightbigg ∇iϕ∇jϕ+ 2gij/parenleftbigg∂ ∂t∇iϕ/parenrightbigg (∇jϕ) = (R−r)|∇ϕ|2+ 2gij∇i(∆ϕ+rϕ−b(t))∇jϕ = (R+r)|∇ϕ|2+ 2gij(∆∇iϕ−Rik∇kϕ)∇jϕ = (R+r)|∇ϕ|2+ ∆|∇ϕ|2−2|∇2ϕ|2−2gijRik∇kϕ∇jϕ = ∆|∇ϕ|2−2|∇2ϕ|2+r|∇ϕ|2, whereRik=1 2Rgikon a surface. On the other hand we may rewrite the evolution equation (5.1. 2) as ∂ ∂t(R−r) = ∆(R−r) + (∆ϕ)2+r(R−r). Then the combination of above two equations yields ∂ ∂th= ∆h−2(|∇2ϕ|2−1 2(∆ϕ)2) +rh = ∆h−2|Mij|2+rh as desired. As a direct consequence of the evolution equation (5.1.5) an d the maximum prin- ciple, we have (5.1.6) R≤C1ert+r for some positive constant C1depending only on the initial metric. On the other hand, it follows from (5.1.2) that Rmin(t) = min x∈MR(x,t) satisﬁes d dtRmin≥Rmin(Rmin−r)≥0 wheneverRmin≤0. This says that (5.1.7) Rmin(t)≥ −C2,for allt>0 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 311 for some positive constant C2depending only on the initial metric. Thus the combination of (5.1.6) and (5.1.7) implies the foll owing long time exis- tence result. Proposition 5.1.3. For any initial metric on a compact surface, the normalized Ricci ﬂow (5.1.1)has a solution for all time. To investigate the long-time behavior of the solution, let u s now divide the dis- cussion into three cases: χ(M)<0;χ(M) = 0; andχ(M)>0. Case(1):χ(M)<0 (i.e.,r<0). From the evolution equation (5.1.2), we have d dtRmin≥Rmin(Rmin−r) ≥r(Rmin−r),onM×[0,+∞) which implies that R−r≥ −˜C1ert,onM×[0,+∞) for some positive constant ˜C1depending only on the initial metric. Thus by combining with (5.1.6) we have (5.1.8) −˜C1ert≤R−r≤C1ert,onM×[0,+∞). Theorem 5.1.4 ( Hamilton [60] ).On a compact surface with χ(M)<0, for any initial metric the solution of the normalized Ricci ﬂow (5.1.1)exists for all time and converges in the C∞topology to a metric with negative constant curvature. Proof. The estimate (5.1.8) shows that the scalar curvature R(x,t) converges exponentially to the negative constant rast→+∞. Fix a tangent vector v∈TxMat a pointx∈Mand let |v|2 t=gij(x,t)vivj. Then we have d dt|v|2 t=/parenleftbigg∂ ∂tgij(x,t)/parenrightbigg vivj = (r−R)|v|2 t which implies /vextendsingle/vextendsingle/vextendsingle/vextendsingled dtlog|v|2 t/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cert,for allt>0 for some positive constant Cdepending only on the initial metric (by using (5.1.8)). Thus|v|2 tconverges uniformly to a continuous function |v|2 ∞ast→+∞and|v|2 ∞/\e}atio\slash= 0 ifv/\e}atio\slash= 0. Since the parallelogram law continues to hold to the limi t, the limiting norm|v|2 ∞comes from an inner product gij(∞). This says, the metrics gij(t) are all equivalent and as t→+∞, the metric gij(t) converges uniformly to a positive-deﬁnite metric tensor gij(∞) which is continuous and equivalent to the initial metric. By the virtue of Shi’s derivative estimates of the unnormali zed Ricci ﬂow in Section 1.4, we see that all derivatives and higher order der ivatives of the curvature of the solution gijof the normalized ﬂow are uniformly bounded on M×[0,+∞). 312 H.-D. CAO AND X.-P. ZHU This shows that the limiting metric gij(∞) is a smooth metric with negative constant curvature and the solution gij(t) converges to the limiting metric gij(∞) in theC∞ topology as t→+∞. Case(2):χ(M) = 0, i.e.,r= 0. From (5.1.6) and (5.1.7) we know that the curvature remains b ounded above and below. To get the convergence, we consider the potential fun ctionϕof (5.1.3) again. The evolution of ϕis given by (5.1.4). We renormalize the function ϕby ˜ϕ(x,t) =ϕ(x,t) +/integraldisplay b(t)dt, onM×[0,+∞). Then, since r= 0, ˜ϕevolves by (5.1.9)∂˜ϕ ∂t= ∆˜ϕ, onM×[0,+∞). From the proof of Lemma 5.1.2, we get (5.1.10)∂ ∂t|∇˜ϕ|2= ∆|∇˜ϕ|2−2|∇2˜ϕ|2. Clearly, we have (5.1.11)∂ ∂t˜ϕ2= ∆˜ϕ2−2|∇˜ϕ|2. Thus it follows that ∂ ∂t(t|∇˜ϕ|2+ ˜ϕ2)≤∆(t|∇˜ϕ|2+ ˜ϕ2). Hence by applying the maximum principle, there exists a posi tive constant C3de- pending only on the initial metric such that (5.1.12) |∇˜ϕ|2(x,t)≤C3 1 +t,onM×[0,+∞). In the following we will use this decay estimate to obtain a de cay estimate for the scalar curvature. By the evolution equations (5.1.2) and (5.1.10), we have ∂ ∂t(R+ 2|∇˜ϕ|2) = ∆(R+ 2|∇˜ϕ|2) +R2−4|∇2˜ϕ|2 ≤∆(R+ 2|∇˜ϕ|2)−R2 sinceR2= (∆˜ϕ)2≤2|∇2˜ϕ|2. Thus by using (5.1.12), we have ∂ ∂t[t(R+ 2|∇˜ϕ|2)] ≤∆[t(R+ 2|∇˜ϕ|2)]−tR2+R+ 2|∇˜ϕ|2 ≤∆[t(R+ 2|∇˜ϕ|2)]−t(R+ 2|∇˜ϕ|2)2+ (1 + 4t|∇˜ϕ|2)(R+ 2|∇˜ϕ|2) ≤∆[t(R+ 2|∇˜ϕ|2)]−[t(R+ 2|∇˜ϕ|2)−(1 + 4C3)](R+ 2|∇˜ϕ|2) ≤∆[t(R+ 2|∇˜ϕ|2)] THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 313 wherevert(R+ 2|∇˜ϕ|2)≥(1 + 4C3). Hence by the maximum principle, there holds (5.1.13) R+ 2|∇˜ϕ|2≤C4 1 +t,onM×[0,+∞) for some positive constant C4depending only on the initial metric. On the other hand, the scalar curvature satisﬁes ∂R ∂t= ∆R+R2,onM×[0,+∞). It is not hard to see that (5.1.14) R≥Rmin(0) 1−Rmin(0)t,onM×[0,+∞), by using the maximum principle. So we obtain the decay estima te for the scalar curvature (5.1.15) |R(x,t)| ≤C5 1 +t,onM×[0,+∞), for some positive constant C5depending only on the initial metric. Theorem 5.1.5 ( Hamilton [60] ).On a compact surface with χ(M) = 0, for any initial metric the solution of the normalized Ricci ﬂow (5.1.1)exists for all time and converges in C∞topology to a ﬂat metric. Proof. Since∂˜ϕ ∂t= ∆˜ϕ, it follows from the maximum principle that |˜ϕ(x,t)| ≤C6,onM×[0,+∞) for some positive constant C6depending only on the initial metric. Recall that ∆˜ ϕ= R. We thus obtain for any tangent vector v∈TxMat a pointx∈M, d dt|v|2 t=/parenleftbigg∂ ∂tgij(x,t)/parenrightbigg vivj =−R(x,t)|v|2 t and then /vextendsingle/vextendsingle/vextendsingle/vextendsinglelog|v|2 t |v|2 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0d dtlog/vextendsingle/vextendsingle/vextendsingle/vextendsinglevt|2dt| =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0R(x,t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle =|˜ϕ(x,t)−˜ϕ(x,0)| ≤2C6, for allx∈Mandt∈[0,+∞).This shows that the solution gij(t) of the normalized Ricci ﬂow are all equivalent. This gives us control of the dia meter and injectivity radius. As before, by Shi’s derivative estimates of the unnormalize d Ricci ﬂow, all deriva- tives and higher order derivatives of the curvature of the so lutiongijof the normalized Ricci ﬂow (5.1.1) are uniformly bounded on M×[0,+∞). By the virtue of Hamilton’s 314 H.-D. CAO AND X.-P. ZHU compactness theorem (Theorem 4.1.5) we see that the solutio ngij(t) subsequentially converges in C∞topology. The decay estimate (5.1.15) implies that each lim it must be a ﬂat metric on M. Clearly, we will ﬁnish the proof if we can show that limit is unique. Note that the solution gij(t) is changing conformally under the Ricci ﬂow (5.1.1) on surfaces. Thus each limit must be conformal to the initial metric, denoted by ¯ gij. Let us denote gij(∞) =eu¯gijto be a limiting metric. Since gij(∞) is ﬂat, it is easy to compute 0 =e−u(¯R−¯∆u),onM, where ¯Ris the curvature of ¯ gijand¯∆ is the Laplacian in the metric ¯ gij. The solution of Poission equation ¯∆u=¯R, onM is unique up to constant. Moreover the constant must be also u niquely determined since the area of the solution of the normalized Ricci ﬂow (5. 1.1) is constant in time. So the limit is unique and we complete the proof of Theorem 5.1 .5. Case(3):χ(M)>0, i.e.,r>0. This is the most diﬃcult case. There exist several proofs by n ow but, in contrast to the previous two cases, none of them depend only on the maxi mum principle type of argument. In fact, all the proofs rely on some combination of the maximum principle argument and certain integral estimate of the curvature. In the pioneer work [60], Hamilton introduced an integral quantity E=/integraldisplay MRlogRdV, which he calls entropy , for the (normalized) Ricci ﬂow on a surface Mwith posi- tive curvature, and showed that the entropy is monotone decr easing under the ﬂow. By combining this entropy estimate with the Harnack inequal ity for the curvature (Corollary 2.5.3), Hamilton obtained the uniform bound on t he curvature of the nor- malized Ricci ﬂow on Mwith positive curvature. Furthermore, he showed that the evolving metric converges to a shrinking Ricci soliton on Mand that the shrinking Ricci soliton must be a round metric on the 2-sphere S2. Subsequently, Chow [36] extended Hamilton’s work to the general case when the curvat ure may change signs. More precisely, he proved that given any initial metric on a c ompact surface Mwith χ(M)>0, the evolving metric under the (normalized) Ricci ﬂow will have positive curvature after a ﬁnite time. Hence, when combined with Hami lton’s result, we know the evolving metric on Mconverges to the round metric on S2. In the following we present a new argument by combining the Li -Yau-Hamilton inequality of the curvature with Perelman’s no local collap sing theorem I′, as was done in the recent joint work of Bing-Long Chen and the author s [15] where they con- sidered the K¨ ahler-Ricci ﬂow on higher dimensional K¨ ahle r manifolds of nonnegative holomorphic bisectional curvature (see [15] for more detai ls). (There are also other proofs for Case (3) by Bartz-Struwe-Ye [6] and Struwe [121]. ) Given any initial metric on Mwithχ(M)>0, we consider the solution gij(t) of the normalized Ricci ﬂow (5.1.1). Recall that the (scalar) c urvatureRsatisﬁes the evolution equation ∂ ∂tR= ∆R+R2−rR. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 315 The corresponding ODE is (5.1.16)ds dt=s2−rs. Let us choose c >1 and close to 1 so that r/(1−c)<minx∈MR(x,0). It is clear that the function s(t) =r/(1−cert)<0 is a solution of the ODE (5.1.16) with s(0)<min x∈MR(x,0). Then the diﬀerence of Randsevolves by (5.1.17)∂ ∂t(R−s) = ∆(R−s) + (R−r+s)(R−s). Since min x∈MR(x,0)−s(0)>0, the maximum principle implies that R−s>0 for all times. We ﬁrst extend the Li-Yau-Hamilton inequality (Theorem 2.5 .2) to the normalized Ricci ﬂow whose curvature may change signs. As in the proof of Theorem 2.5.2, we consider the quantity L= log(R−s). It is easy to compute ∂L ∂t= ∆L+|∇L|2+R−r+s. Then we set Q=∂L ∂t− |∇L|2−s= ∆L+R−r. By a direct computation and using the estimate (5.1.8), we ha ve ∂ ∂tQ= ∆/parenleftbigg∂L ∂t/parenrightbigg + (R−r)∆L+∂R ∂t = ∆Q+ 2|∇2L|2+ 2/a\}b∇acketle{t∇L,∇(∆L)/a\}b∇acket∇i}ht+R|∇L|2 + (R−r)∆L+ ∆R+R(R−r) = ∆Q+ 2|∇2L|2+ 2/a\}b∇acketle{t∇L,∇Q/a\}b∇acket∇i}ht+ 2(R−r)∆L+ (R−r)2 + (r−s)∆L+s|∇L|2+r(R−r) = ∆Q+ 2/a\}b∇acketle{t∇L,∇Q/a\}b∇acket∇i}ht+ 2|∇2L|2+ 2(R−r)∆L+ (R−r)2 + (r−s)Q+s|∇L|2+s(R−r) ≥∆Q+ 2/a\}b∇acketle{t∇L,∇Q/a\}b∇acket∇i}ht+Q2+ (r−s)Q+s|∇L|2−C. Here and below Cis denoted by various positive constants depending only on t he initial metric. In order to control the bad term s|∇L|2, we consider ∂ ∂t(sL) = ∆(sL) +s|∇L|2+s(R−r+s) +s(s−r)L ≥∆(sL) + 2/a\}b∇acketle{t∇L,∇(sL)/a\}b∇acket∇i}ht −s|∇L|2−C 316 H.-D. CAO AND X.-P. ZHU by using the estimate (5.1.8) again. Thus ∂ ∂t(Q+sL)≥∆(Q+sL) + 2/a\}b∇acketle{t∇L,∇(Q+sL)/a\}b∇acket∇i}ht+Q2+ (r−s)Q−C ≥∆(Q+sL) + 2/a\}b∇acketle{t∇L,∇(Q+sL)/a\}b∇acket∇i}ht+1 2[(Q+sL)2−C2], sincesLis bounded by (5.1.8). This, by the maximum principle, impli es that Q≥ −C, for allt∈[0,+∞). Then for any two points x1,x2∈Mand two times t2> t1≥0, and a path γ: [t1,t2]→Mconnecting x1tox2, we have L(x2,t2)−L(x1,t1) =/integraldisplayt2 t1d dtL(γ(t),t)dt =/integraldisplayt2 t1/parenleftbigg∂L ∂t+/a\}b∇acketle{t∇L,˙γ/a\}b∇acket∇i}ht/parenrightbigg dt ≥ −1 4∆−C(t2−t1) where ∆ = ∆(x1,t1;x2,t2) = inf/braceleftbigg/integraldisplayt2 t1|˙γ(t)|2 gij(t)dt|γ: [t1,t2]→Mwithγ(t1) =x1,γ(t2) =x2/bracerightbigg . Thus we have proved the following Harnack inequality. Lemma 5.1.6 ( Chow [36] ).There exists a positive constant C depending only on the initial metric such that for any x1,x2∈Mandt2>t1≥0, R(x1,t1)−s(t1)≤e∆ 4+C(t2−t1)(R(x2,t2)−s(t2)) where ∆ = inf/braceleftbigg/integraldisplayt2 t1|˙γ(t)|2 tdt|γ: [t1,t2]→M withγ (t1) =x1,γ(t2) =x2/bracerightbigg . We now state and prove the uniform bound estimate for the curv ature. Proposition 5.1.7. Let(M,g ij(t))be a solution of the normalized Ricci ﬂow on a compact surface with χ(M)>0. Then there exist a time t0>0and a positive constantCsuch that the estimate C−1≤R(x,t)≤C holds for all x∈Mandt∈[t0,+∞). Proof. Recall that R(x,t)≥s(t) =r 1−cert,onM×[0,+∞). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 317 For anyε∈(0,r), there exists a large enough t0>0 such that (5.1.18) R(x,t)≥ −ε2,onM×[t0,+∞). Lettbe any ﬁxed time with t≥t0+ 1. Obviously there is some point x0∈Msuch thatR(x0,t+ 1) =r. Consider the geodesic ball Bt(x0,1), centered at x0and radius 1 with respect to the metric at the ﬁxed time t. For any point x∈Bt(x0,1), we choose a geodesic γ: [t,t+ 1]→Mconnecting xandx0with respect to the metric at the ﬁxed time t. Since ∂ ∂tgij= (r−R)gij≤2rgij onM×[t0,+∞), we have /integraldisplayt+1 t|˙γ(τ)|2 τdτ≤e2r/integraldisplayt+1 t|˙γ(τ)|2 tdτ≤e2r. Then by Lemma 5.1.6, we have R(x,t)≤s(t) + exp/braceleftbigg1 4e2r+C/bracerightbigg ·(R(x0,t+ 1)−s(t+ 1)) (5.1.19) ≤C1,asx∈Bt(x0,1), for some positive constant C1depending only on the initial metric. Note the the corresponding unnormalized Ricci ﬂow in this case has ﬁnite maximal time since its volume decreases at a ﬁxed rate −4πχ(M)<0. Hence the no local collapsing theorem I′(Theorem 3.3.3) implies that the volume of Bt(x0,1) with respect to the metric at the ﬁxed time tis bounded from below by (5.1.20) Vol t(Bt(x0,1))≥C2 for some positive constant C2depending only on the initial metric. We now want to bound the diameter of ( M,g ij(t)) from above. The following argument is analogous to Yau in [128] where he got a lower boun d for the volume of geodesic balls of a complete Riemannian manifold with nonne gative Ricci curvature. Without loss of generality, we may assume that the diameter o f (M,g ij(t)) is at least 3. Choose a point x1∈Msuch that the distance dt(x0,x1) betweenx1andx0with respect to the metric at the ﬁxed time tis at least a half of the diameter of ( M,g ij(t)). By (5.1.18), the standard Laplacian comparison theorem (c. f. [112]) implies ∆ρ2= 2ρ∆ρ+ 2≤2(1 +ερ) + 2 in the sense of distribution, where ρis the distance function from x1(with respect to the metricgij(t)). That is, for any ϕ∈C∞ 0(M),ϕ≥0, we have (5.1.21) −/integraldisplay M∇ρ2· ∇ϕ≤/integraldisplay M[2(1 +ερ) + 2]ϕ. SinceC∞ 0(M) functions can be approximated by Lipschitz functions in th e above inequality, we can set ϕ(x) =ψ(ρ(x)),x∈M, whereψ(s) is given by ψ(s) =  1, 0≤s≤dt(x0,x1)−1, ψ′(s) =−1 2, d t(x0,x1)−1≤s≤dt(x0,x1) + 1, 0, s ≥dt(x0,x1) + 1. 318 H.-D. CAO AND X.-P. ZHU Thus, by using (5.1.20), the left hand side of (5.1.21) is −/integraldisplay M∇ρ2· ∇ϕ =/integraldisplay Bt(x1,dt(x0,x1)+1)\Bt(x1,dt(x0,x1)−1)ρ ≥(dt(x0,x1)−1)Vol t(Bt(x1,dt(x0,x1) + 1) \Bt(x1,dt(x0,x1)−1)) ≥(dt(x0,x1)−1)Vol t(Bt(x0,1)) ≥(dt(x0,x1)−1)C2, and the right hand side of (5.1.21) is /integraldisplay M[2(1 +ερ) + 2]ϕ≤/integraldisplay Bt(x1,dt(x0,x1)+1)[2(1 +ερ) + 2] ≤[2(1 +εdt(x0,x1)) + 4]Vol t(Bt(x1,dt(x0,x1) + 1)) ≤[2(1 +εdt(x0,x1)) + 4]A whereAis the area of Mwith respect to the initial metric. Here we have used the fact that the area of solution of the normalized Ricci ﬂow is c onstant in time. Hence C2(dt(x0,x1)−1)≤[2(1 +εdt(x0,x1)) + 4]A, which implies, by choosing ε>0 small enough, dt(x0,x1)≤C3 for some positive constant C3depending only on the initial metric. Therefore, the diameter of ( M,g ij(t)) is uniformly bounded above by (5.1.22) diam( M,g ij(t))≤2C3 for allt∈[t0,+∞). We then argue, as in deriving (5.1.19), by applying Lemma 5.1 .6 again to obtain R(x,t)≤C4,onM×[t0,+∞) for some positive constant C4depending only on the initial metric. It remains to prove a positive lower bound estimate of the cur vature. First, we note that the function s(t)→0 ast→+∞, and the average scalar curvature of the solution equals to r, a positive constant. Thus the Harnack inequality in Lemma 5 .1.6 and the diameter estimate (5.1.22) imply a positive lower bo und for the curvature. Therefore we have completed the proof of Proposition 5.1.7. Next we consider long-time convergence of the normalized ﬂo w. Recall that the trace-free part of the Hessian of the potenti alϕof the curvature is the tensor Mijdeﬁned by Mij=∇i∇jϕ−1 2∆ϕ·gij, where by (5.1.3), ∆ϕ=R−r. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 319 Lemma 5.1.8. We have (5.1.23)∂ ∂t|Mij|2= ∆|Mij|2−2|∇kMij|2−2R|Mij|2,onM×[0,+∞). Proof. First we note the time-derivative of the Levi-Civita connec tion is ∂ ∂tΓk ij=1 2gkl/parenleftbigg ∇i∂ ∂tgjl+∇j∂ ∂tgil− ∇ l∂ ∂tgij/parenrightbigg =1 2/parenleftbig −∇iR·δk j− ∇ jR·δk i+∇kR·gij/parenrightbig . By using this and (5.1.4), we have ∂ ∂tMij=∇i∇j/parenleftbigg∂ϕ ∂t/parenrightbigg −/parenleftbigg∂ ∂tΓk ij/parenrightbigg ∇kϕ−1 2∂ ∂t[(R−r)gij] =∇i∇j∆ϕ+1 2(∇iR· ∇jϕ+∇jR· ∇iϕ− /a\}b∇acketle{t∇R,∇ϕ/a\}b∇acket∇i}htgij) −1 2∆R·gij+rMij. Since on a surface, Rijkl=1 2R(gilgjk−gikgjl), we have ∇i∇j∆ϕ =∇i∇k∇j∇kϕ− ∇ i(Rjl∇lϕ) =∇k∇i∇j∇kϕ−Rl ikj∇l∇kϕ−Ril∇j∇lϕ−Rjl∇i∇lϕ− ∇ iRjl∇lϕ = ∆∇i∇jϕ− ∇k(Rl ikj∇lϕ)−Rl ikj∇l∇kϕ −Ril∇j∇lϕ−Rjl∇i∇lϕ− ∇ iRjl∇lϕ = ∆∇i∇jϕ−1 2(∇iR· ∇jϕ+∇jR· ∇iϕ− /a\}b∇acketle{t∇R,∇ϕ/a\}b∇acket∇i}htgij) −2R/parenleftbigg ∇i∇jϕ−1 2∆ϕ·gij/parenrightbigg . Combining these identities, we get ∂ ∂tMij= ∆∇i∇jϕ−1 2∆R·gij+ (r−2R)Mij = ∆/parenleftbigg ∇i∇jϕ−1 2(R−r)gij/parenrightbigg + (r−2R)Mij. Thus the evolution Mijis given by (5.1.24)∂Mij ∂t= ∆Mij+ (r−2R)Mij. Now the lemma follows from (5.1.24) and a straightforward co mputation. 320 H.-D. CAO AND X.-P. ZHU Proposition 5.1.7 tells us that the curvature Rof the solution to the normalized Ricci ﬂow is uniformly bounded from below by a positive const ant fortlarge. Thus we can apply the maximum principle to the equation (5.1.23) in L emma 5.1.8 to obtain the following estimate. Proposition 5.1.9. Let(M,g ij(t))be a solution of the normalized Ricci ﬂow on a compact surface with χ(M)>0. Then there exist positive constants candC depending only on the initial metric such that |Mij|2≤Ce−ct,onM×[0,+∞). Now we consider a modiﬁcation of the normalized Ricci ﬂow. Co nsider the equa- tion (5.1.25)∂ ∂tgij= 2Mij= (r−R)gij+ 2∇i∇jϕ. As we saw in Section 1.3, the solution of this modiﬁed ﬂow diﬀe rs from that of the normalized Ricci ﬂow only by a one parameter family of diﬀeom orphisms generated by the gradient vector ﬁeld of the potential function ϕ. Since the quantity |Mij|2is invariant under diﬀeomorphisms, the estimate |Mij|2≤Ce−ctalso holds for the solu- tion of the modiﬁed ﬂow (5.1.25). This exponential decay est imate then implies the solutiongij(x,t) of the modiﬁed ﬂow (5.1.25) converges exponentially to a co ntinuous metricgij(∞) ast→+∞. Furthermore, by the virtue of Hamilton’s compactness theorem (Theorem 4.1.5) we see that the solution gij(x,t) of the modiﬁed ﬂow actu- ally converges exponentially in C∞topology to gij(∞). Moreover the limiting metric gij(∞) satisﬁes Mij= (r−R)gij+ 2∇i∇jϕ= 0,onM. That is, the limiting metric is a shrinking gradient Ricci so liton on the surface M. The next result was ﬁrst obtained by Hamilton in [60]. The fol lowing simpliﬁed proof by using the Kazdan-Warner identity was widely known t o experts in the ﬁeld. Proposition 5.1.10. On a compact surface there are no shrinking Ricci solitons other than constant curvature. Proof. By deﬁnition, a shrinking Ricci soliton on a compact surface Mis given by (5.1.26) ∇iXj+∇jXi= (R−r)gij for some vector ﬁeld X=Xj. By contracting the above equation by Rg−1, we have 2R(R−r) = 2RdivX, and hence /integraldisplay M(R−r)2dV=/integraldisplay MR(R−r)dV=/integraldisplay MRdivXdV. SinceXis a conformal vector ﬁeld (by the Ricci soliton equation (5. 1.26)), by inte- grating by parts and applying the Kazdan-Warner identity [7 7], we obtain /integraldisplay M(R−r)2dV=−/integraldisplay M/a\}b∇acketle{t∇R,X/a\}b∇acket∇i}htdV= 0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 321 HenceR≡r, and the lemma is proved. Now back to the solution of the modiﬁed ﬂow (5.1.25). We have s een the curvature converges exponentially to its limiting value in the C∞topology. But since there are no nontrivial soliton on M, we must have Rconverging exponentially to the constant valuerin theC∞topology. This then implies that the unmodiﬁed ﬂow (5.1.1) w ill converge to a metric of positive constant curvature in the C∞topology. In conclusion, we have proved the following main theorem of t his section. Theorem 5.1.11 ( Hamilton [60], Chow [36] ).On a compact surface with χ(M)> 0, for any initial metric, the solution of the normalized Ricc i ﬂow (5.1.1)exists for all time, and converges in the C∞topology to a metric with positive constant curvature. 5.2. Diﬀerentiable Sphere Theorems in 3-D and 4-D. An important prob- lem in Riemannian geometry is to understand the inﬂuence of c urvatures, in particular the sign of curvatures, on the topology of underlying manifo lds. Classical results of this type include sphere theorem and its reﬁnements stated b elow (see, for example, Theorem 6.1 and Theorem 7.16 in Cheeger-Ebin [22], and Theor em 6.6 in Cheeger- Ebin [22]). In this section we shall use the long-time behavi or of the Ricci ﬂow on positively curved manifolds to establish Hamilton’s diﬀer entiable sphere theorems in dimensions three and four. Let us ﬁrst recall the classical sphere theorems. Given a Rem annian manifold M, we denote by KMthe sectional curvature of M. Classical Sphere Theorems. LetMbe a complete, simply connected n- dimensional manifold. (i)If1 4<K M≤1, thenMis homeomorphic to the n-sphere Sn. (ii)There exists a positive constant δ∈(1 4,1)such that if δ <K M≤1, thenM is diﬀeomorphic to the n-sphere Sn. Result (ii) is called the diﬀerentiable sphere theorem. If w e relax the assumptions on the strict lower bound in (i), then we have the following ri gidity result. Berger’s Rigidity Theorem. LetMbe a complete, simply connected n- dimensional manifold with1 4≤KM≤1. Then either Mis homeomorphic to Snor Mis isometric to a symmetric space. We remark that it follows from the classiﬁcation of symmetri c spaces (see for example [68]) that the only simply connected symmetric spac es with positive curvature areSn,CPn 2,QPn 4, and the Cayley plane. In early and mid 80’s respectively, Hamilton [58], [59] used the Ricci ﬂow to prove the following diﬀerential sphere theorems. Theorem 5.2.1 ( Hamilton [58] ).A compact three-manifold with positive Ricci curvature must be diﬀeomorphic to the three-sphere S3or a quotient of it by a ﬁnite group of ﬁxed point free isometries in the standard metric. Theorem 5.2.2 ( Hamilton [59] ).A compact four-manifold with positive curva- ture operator is diﬀeomorphic to the four-sphere S4or the real projective space RP4. Note that in above two theorems, we only assume curvatures to be strictly pos- itive, but not any strong pinching conditions as in the class ical sphere theorems. In fact, one of the important special features discovered by Ha milton is that if the ini- tial metric has positive curvature, then the metric will get rounder and rounder as it 322 H.-D. CAO AND X.-P. ZHU evolves under the Ricci ﬂow, at least in dimension three and f our, so any small initial pinching will get improved. Indeed, the pinching estimate i s a key step in proving both Theorem 5.2.1 and 5.2.2. The following results are concerned with compact three-man ifolds or four- manifolds with weakly positive curvatures. Theorem 5.2.3 ( Hamilton [59] ). (i) A compact three-manifold with nonnegative Ricci curvat ure is diﬀeomorphic toS3, or a quotient of one of the spaces S3orS2×R1orR3by a group of ﬁxed point free isometries in the standard metrics. (ii) A compact four-manifold with nonnegative curvature op erator is diﬀeomor- phic to S4orCP2orS2×S2, or a quotient of one of the spaces S4orCP2or S3×R1orS2×S2orS2×R2orR4by a group of ﬁxed point free isometries in the standard metrics. The rest of the section will be devoted to prove Theorems 5.2. 1-5.2.3. Recall that the curvature operator Mαβevolves by (5.2.1)∂ ∂tMαβ= ∆Mαβ+M2 αβ+M# αβ. where (see Section 1.3 and Section 2.4) M2 αβis the operator square M2 αβ=MαγMβγ andM# αβis the Lie algebra so(n) square M# αβ=Cγζ αCηθ βMγηMζθ. We begin with the curvature pinching estimates of the Ricci ﬂ ow in three dimen- sions. In dimension n= 3, we know that M# αβis the adjoint matrix of Mαβ. If we diagonalize Mαβwith eigenvalues λ≥µ≥νso that (Mαβ) = λ µ ν , thenM2 αβandM# αβare also diagonal, with (M2 αβ) = λ2 µ2 ν2 and (M# αβ) = µν λν λµ , and the ODE corresponding to PDE (5.2.1) is then given by the s ystem (5.2.2)  d dtλ=λ2+µν, d dtµ=µ2+λν, d dtν=ν2+λµ. Lemma 5.2.4. For anyε∈[0,1 3], the pinching condition Rij≥0and R ij≥εRgij THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 323 is preserved by the Ricci ﬂow. Proof. If we diagonalize the 3 ×3 curvature operator matrix Mαβwith eigen- valuesλ≥µ≥ν, then nonnegative sectional curvature corresponds to ν≥0 and nonnegative Ricci curvature corresponds to the inequality µ+ν≥0. Also, the scalar curvatureR=λ+µ+ν. So we need to show µ+ν≥0 andµ+ν≥δλ,withδ= 2ε/(1−2ε), are preserved by the Ricci ﬂow. By Hamilton’s advanced maxim um principle (Theo- rem 2.3.1), it suﬃces to show that the closed convex set K={Mαβ|µ+ν≥0 andµ+ν≥δλ} is preserved by the ODE system (5.2.2). Now suppose we have diagonalized Mαβwith eigenvalues λ≥µ≥νatt= 0, then bothM2 αβandM# αβare diagonal, so the matrix Mαβremains diagonal for t>0. Moreover, since d dt(µ−ν) = (µ−ν)(µ+ν−λ), it is clear that µ≥νfort >0 also. Similarly, we have λ≥µfort >0. Hence the inequalities λ≥µ≥νpersist. This says that the solutions of the ODE system (5.2. 2) agree with the original choice for the eigenvalues of the cur vature operator. The condition µ+ν≥0 is clearly preserved by the ODE, because d dt(µ+ν) =µ2+ν2+λ(µ+ν)≥0. It remains to check d dt(µ+ν)≥δd dtλ or µ2+λν+ν2+λµ≥δ(λ2+µν) on the boundary where µ+ν=δλ≥0. In fact, since (λ−ν)µ2+ (λ−µ)ν2≥0, we have λ(µ2+ν2)≥(µ+ν)µν. Hence µ2+µλ+ν2+νλ≥/parenleftbiggµ+ν λ/parenrightbigg (λ2+µν) =δ(λ2+µν). 324 H.-D. CAO AND X.-P. ZHU So we get the desired pinching estimate. Proposition 5.2.5. Suppose that the initial metric of the solution to the Ricci ﬂow onM3×[0,T)has positive Ricci curvature. Then for any ε >0we can ﬁnd Cε<+∞such that /vextendsingle/vextendsingle/vextendsingle/vextendsingleRij−1 3Rgij/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤εR+Cε for all subsequent t∈[0,T). Proof. Again we consider the ODE system (5.2.2). Let Mαβbe diagonalized with eigenvalues λ≥µ≥νatt= 0. We saw in the proof of Lemma 5.2.4 the inequalities λ≥µ≥νpersist fort>0. We only need to show that there are positive constants δandCsuch that the closed convex set K={Mαβ|λ−ν≤C(λ+µ+ν)1−δ} is preserved by the ODE. We compute d dt(λ−ν) = (λ−ν)(λ+ν−µ) and d dt(λ+µ+ν) = (λ+µ+ν)(λ+ν−µ) +µ2 +µ(µ+ν) +λ(µ−ν) ≥(λ+µ+ν)(λ+ν−µ) +µ2. Thus, without loss of generality, we may assume λ−ν >0 and get d dtlog(λ−ν) =λ+ν−µ and d dtlog(λ+µ+ν)≥λ+ν−µ+µ2 λ+µ+ν. By Lemma 5.2.4, there exists a positive constant Cdepending only on the initial metric such that λ≤λ+µ≤C(µ+ν)≤2Cµ, λ+ν−µ≤λ+µ+ν≤6Cµ, and hence with ǫ= 1/36C2, d dtlog(λ+µ+ν)≥(1 +ǫ)(λ+ν−µ). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 325 Therefore with (1 −δ) = 1/(1 +ǫ), d dtlog((λ−ν)/(λ+µ+ν)1−δ)≤0. This proves the proposition. We now are ready to prove Theorem 5.2.1. Proof of Theorem 5.2.1. LetMbe a compact three-manifold with positive Ricci curvature and let the metric evolve by the Ricci ﬂow. By Lemma 5.2.4 we know that there exists a positive constant β >0 such that Rij≥βRg ij for allt≥0 as long as the solution exists. The scalar curvature evolve s by ∂R ∂t= ∆R+ 2|Rij|2 ≥∆R+2 3R2, which implies, by the maximum principle, that the scalar cur vature remains positive and tends to + ∞in ﬁnite time. We now use a blow up argument as in Section 4.3 to get the follow ing gradient estimate. Claim. For anyε>0, there exists a positive constant Cε<+∞such that for any timeτ≥0, we have max t≤τmax x∈M|∇Rm(x,t)| ≤εmax t≤τmax x∈M|Rm(x,t)|3 2+Cε. We argue by contradiction. Suppose the above gradient estim ate fails for some ﬁxedε0>0. Pick a sequence Cj→+∞, and pick points xj∈Mand timesτjsuch that |∇Rm(xj,τj)| ≥ε0max t≤τjmax x∈M|Rm(x,t)|3 2+Cj, j= 1,2,.... Choosexjto be the origin, and pull the metric back to a small ball on the tangent spaceTxjMof radiusrjproportional to the reciprocal of the square root of the maximum curvature up to time τj(i.e., max t≤τjmax x∈M|Rm(x, t)|). Clearly the maximum curvatures go to inﬁnity by Shi’s derivative estima te of curvature (Theorem 1.4.1). Dilate the metrics so that the maximum curvature max t≤τjmax x∈M|Rm(x,t)| becomes 1 and translate time so that τjbecomes the time 0. By Theorem 4.1.5, we can take a (local) limit. The limit metric satisﬁes |∇Rm(0,0)| ≥ε0>0. However the pinching estimate in Proposition 5.2.5 tells us the limit metric has Rij−1 3Rgij≡0. 326 H.-D. CAO AND X.-P. ZHU By using the contracted second Bianchi identity 1 2∇iR=∇jRij=∇j/parenleftbigg Rij−1 3Rgij/parenrightbigg +1 3∇iR, we get ∇iR≡0 and then ∇iRjk≡0. For a three-manifold, this in turn implies ∇Rm= 0 which is a contradiction. Hence we have proved the gradient e stimate claimed. We can now show that the solution to the Ricci ﬂow becomes roun d as the time ttends to the maximal time T. We have seen that the scalar curvature goes to inﬁnity in ﬁnite time. Pick a sequence of points xj∈Mand timesτjwhere the curvature at xjis as large as it has been anywhere for 0 ≤t≤τjandτjtends to the maximal time. Since |∇Rm|is very small compared to |Rm(xj,τj)|by the above gradient estimate and |Rij−1 3Rgij|is also very small compared to |Rm(xj,τj)|by Proposition 5.2.5, the curvature is nearly constant and pos itive in a large ball around xjat the time τj. But then the Bonnet-Myers’ theorem tells us this is the whol e manifold. For jlarge enough, the sectional curvature of the solution at the timeτjis suﬃciently pinched. Then it follows from the Klingenberg in jectivity radius estimate (see Section 4.2) that the injectivity radius of the metric a t timeτjis bounded from below byc//radicalbig |Rm(xj,τj)|for some positive constant cindependent of j. Dilate the metrics so that the maximum curvature |Rm(xj,τj)|= max t≤τjmax x∈M|Rm(x,t)| becomes 1 and shift the time τjto the new time 0. Then we can apply Hamilton’s compactness theorem (Theorem 4.1.5) to take a limit. By the p inching estimate in Proposition 5.2.5, we know that the limit has positive const ant curvature which is either the round S3or a metric quotient of the round S3. Consequently, the compact three-manifold Mis diﬀeomorphic to the round S3or a metric quotient of the round S3. Next we consider the pinching estimates of the Ricci ﬂow on a c ompact four- manifoldMwith positive curvature operator. In dimension 4, we saw in Section 1.3 when we decompose orthog onally Λ2= Λ2 +⊕Λ2 −into the eigenspaces of Hodge star with eigenvalue ±1, we have a block decomposition of Mαβas Mαβ=/parenleftbiggA B tB C/parenrightbigg and then M# αβ= 2/parenleftbiggA#B# tB#C#/parenrightbigg whereA#,B#,C#are the adjoints of 3 ×3 submatrices as before. Thus the ODE d dtMαβ=M2 αβ+M# αβ THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 327 corresponding to the PDE (5.2.1) breaks up into the system of three equations (5.2.3)  d dtA=A2+BtB+ 2A#, d dtB=AB+BC+ 2B#, d dtC=C2+tBB+ 2C#. As shown in Section 1.3, by the Bianchi identity, we know that trA= trC.For the symmetric matrices AandC, we can choose an orthonormal basis x1,x2,x3of Λ2 + such that A= a10 0 0a20 0 0a3 , and an orthonormal basis z1,z2,z3of Λ2 −such that C= c10 0 0c20 0 0c3 . For matrix B, we can choose orthonormal basis y+ 1,y+ 2,y+ 3of Λ2 +andy− 1,y− 2, y− 3of Λ2 −such that B= b10 0 0b20 0 0b3 . with 0 ≤b1≤b2≤b3. We may also arrange the eigenvalues of AandCasa1≤a2≤ a3andc1≤c2≤c3. In view of the advanced maximum principle Theorem 2.3.1, we only need to establish the pinching estimates for the ODE (5. 2.3). Note that a1= inf{A(x,x)|x∈Λ2 +and|x|= 1}, a3= sup{A(x,x)|x∈Λ2 +and|x|= 1}, c1= inf{C(z,z)|z∈Λ2 −and|z|= 1}, c3= sup{C(z,z)|z∈Λ2 −and|z|= 1}. We can compute their derivatives by Lemma 2.3.3 as follows: (5.2.4)  d dta1≥a2 1+b2 1+ 2a2a3, d dta3≤a2 3+b2 3+ 2a1a2, d dtc1≥c2 1+b2 1+ 2c2c3, d dtc3≤c2 3+b2 3+ 2c1c2. 328 H.-D. CAO AND X.-P. ZHU We shall make the pinching estimates by using the functions b2+b3anda−2b+c, wherea=a1+a2+a3=c=c1+c2+c3andb=b1+b2+b3. Since b2+b3=B(y+ 2,y− 2) +B(y+ 3,y− 3) = sup{B(y+,y−) +B(˜y+,˜y−)|y+,˜y+∈Λ2 +with|y+|=|˜y+|= 1, y+⊥˜y+,andy−,˜y−∈Λ2 −with|y−|=|˜y−|= 1,y−⊥˜y−}, We compute by Lemma 2.3.3, d dt(b2+b3)≤d dtB(y+ 2,y− 2) +d dtB(y+ 3,y− 3) (5.2.5) =AB(y+ 2,y− 2) +BC(y+ 2,y− 2) + 2B#(y+ 2,y− 2) +AB(y+ 3,y− 3) +BC(y+ 3,y− 3) + 2B#(y+ 3,y− 3) =b2A(y+ 2,y+ 2) +b2C(y− 2,y− 2) + 2b1b3 +b3A(y+ 3,y+ 3) +b3C(y− 3,y− 3) + 2b1b2 ≤a2b2+a3b3+b2c2+b3c3+ 2b1b2+ 2b1b3, where we used the facts that A(y+ 2,y+ 2) +A(y+ 3,y+ 3)≤a2+a3andC(y− 2,y− 2) + C(y− 3,y− 3)≤c2+c3. Note also that the function a=trA=c=trCis linear, and the function bis given by b=B(y+ 1,y− 1) +B(y+ 2,y− 2) +B(y+ 3,y− 3) = sup/braceleftig B(Ty+ 1,˜Ty− 1) +B(Ty+ 2,˜Ty− 2) +B(Ty+ 3,˜Ty− 3)|T,˜Tare othogonal transformations of Λ2 +and Λ2 −respectively/bracerightig . Indeed, B(Ty+ 1,˜Ty− 1) +B(Ty+ 2,˜Ty− 2) +B(Ty+ 3,˜Ty− 3) =B(y+ 1,T−1˜T(y− 1)) +B(y+ 2,T−1˜T(y− 2)) +B(y+ 3,T−1˜T(y− 3)) =b1t11+b2t22+b3t33 wheret11,t22,t33are diagonal elements of the orthogonal matrix T−1˜Twith t11,t22,t33≤1. Thus by using Lemma 2.3.3 again, we compute d dt(a−2b+c)≥tr/parenleftbiggd dtA−2d dtB+d dtC/parenrightbigg = tr((A−B)2+ (C−B)2+ 2(A#−2B#+C#)) evaluated in those coordinates where Bis diagonal as above. Recalling the deﬁnition of Lie algebra product P#Q=1 2εαβγεζηθPβηQγθ withεαβγbeing the permutation tensor, we see that the Lie algebra pro duct # gives THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 329 a symmetric bilinear operation on matrices, and then tr(2(A#−2B#+C#)) = tr ((A−C)#+ (A+ 2B+C)#(A−2B+C)) =−1 2tr (A−C)2+1 2(tr (A−C))2 + tr ((A+ 2B+C)#(A−2B+C)) =−1 2tr (A−C)2+ tr ((A+ 2B+C)#(A−2B+C)) by the Bianchi identity. It is easy to check that tr (A−B)2+ tr (C−B)2−1 2tr (A−C)2=1 2tr (A−2B+C)2≥0. Thus we obtain d dt(a−2b+c)≥tr((A+ 2B+C)#(A−2B+C)) SinceMαβ≥0 and Mαβ= A B tB C , we see that A+ 2B+C≥0 andA−2B+C≥0, by applying Mαβto the vectors (x,x) and (x,−x). It is then not hard to see tr ((A+ 2B+C)#(A−2B+C))≥(a1+ 2b1+c1)(a−2b+c). Hence we obtain (5.2.6)d dt(a−2b+c)≥(a1+ 2b1+c1)(a−2b+c). We now state and prove the following pinching estimates of Ha milton for the associated ODE (5.2.3). Proposition 5.2.6 ( Hamilton [59] ).If we choose successively positive constants Glarge enough, Hlarge enough, δsmall enough, Jlarge enough, εsmall enough, Klarge enough, θsmall enough, and Llarge enough, with each depending on those chosen before, then the closed convex subset Xof{Mαβ≥0}deﬁned by the inequalities (1)(b2+b3)2≤Ga1c1, (2)a3≤Ha1andc3≤Hc1, (3)(b2+b3)2+δ≤Ja1c1(a−2b+c)δ, (4)(b2+b3)2+ε≤Ka1c1, (5)a3≤a1+La1−θ 1andc3≤c1+Lc1−θ 1, is preserved by ODE (5.2.3). Moreover every compact subset of {Mαβ>0}lies in some such set X. Proof. Clearly the subset Xis closed and convex. We ﬁrst note that we may assumeb2+b3>0 because if b2+b3= 0, then from (5.2.5), b2+b3will remain 330 H.-D. CAO AND X.-P. ZHU zero and then the inequalities (1), (3) and (4) concerning b2+b3are automatically satisﬁed. Likewise we may assume a3>0 andc3>0 from (5.2.4). LetGbe a ﬁxed positive constant. To prove the inequality (1) we on ly need to check (5.2.7)d dtloga1c1 (b2+b3)2≥0 whenever (b2+b3)2=Ga1c1andb2+b3>0. Indeed, it follows from (5.2.4) and (5.2.5) that d dtloga1≥2b1+ 2a3+(a1−b1)2 a1+ 2a3 a1(a2−a1), (5.2.8) d dtlogc1≥2b1+ 2c3+(c1−b1)2 c1+ 2c3 c1(c2−c1), (5.2.9) and d dtlog(b2+b3)≤2b1+a3+c3−b2 b2+b3[(a3−a2) + (c3−c2)], (5.2.10) which immediately give the desired inequality (5.2.7). By (5.2.4), we have (5.2.11)d dtloga3≤a3+ 2a1+b2 3 a3−2a1 a3(a3−a2). From the inequality (1) there holds b2 3≤Ga1c1. Since trA= trC,c1≤c1+c2+c3= a1+a2+a3≤3a3which shows b2 3 a3≤3Ga1. Thus by (5.2.8) and (5.2.11), d dtloga3 a1≤(3G+ 2)a1−a3. So ifH≥(3G+ 2), then the inequalities a3≤Ha1and likewise c3≤Hc1are preserved. For the inequality (3), we compute from (5.2.8)-(5.2.10) d dtloga1c1 (b2+b3)2≥(a1−b1)2 a1+(c1−b1)2 c1+ 2a3 a1(a2−a1) + 2c3 c1(c2−c1) +2b2 b2+b3[(a3−a2) + (c3−c2)]. Ifb1≤a1/2, then (a1−b1)2 a1≥a1 4≥1 4Ha3, and ifb1≥a1/2, then 2b2 b2+b3≥2b2√Ga1c1≥2b2√3Ga1a3≥2b2√ 3GH·a1≥1√ 3GH. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 331 Thus by combining with 3 a3≥c3, we have d dtloga1c1 (b2+b3)2≥δ(a3−a1) +δ(c3−c1) providedδ≤min(1 24H,1√ 3GH). On the other hand, it follows from (5.2.6) and (5.2.10) that d dtlogb2+b3 a−2b+c≤(a3−a1) + (c3−c1). Therefore the inequality (3) (b2+b3)2+δ≤Ja1c1(a−2b+c)δ will be preserved by any positive constant J. To verify the inequality (4), we ﬁrst note that there is a smal lη>0 such that b≤(1−η)a, on the set deﬁned by the inequality (3). Indeed, if b≤a 2, this is trivial and if b≥a 2, then /parenleftiga 3/parenrightig2+δ ≤(b2+b3)2+δ≤2δJa2(a−b)δ which makes b≤(1−η)afor someη>0 small enough. Consequently, ηa≤a−b≤3(a3−b1) which implies either a3−a1≥1 6ηa, or a1−b1≥1 6ηa. Thus as in the proof of the inequality (3), we have d dtloga1c1 (b2+b3)2≥δ(a3−a1) and d dtloga1c1 (b2+b3)2≥(a1−b1)2 a1, which in turn implies d dtloga1c1 (b2+b3)2≥/parenleftbigg max/braceleftbigg1 6ηδ,1 36η2/bracerightbigg/parenrightbigg ·a. On the other hand, it follows from (5.2.10) that d dtlog(b2+b3)≤2b1+a3+c3≤4a 332 H.-D. CAO AND X.-P. ZHU sinceMαβ≥0. Then ifε>0 is small enough d dtloga1c1 (b2+b3)2+ε≥0 and it follows that the inequality (4) is preserved by any pos itiveK. Finally we consider the inequality (5). From (5.2.8) we have d dtloga1≥a1+ 2a3 and then for θ∈(0,1), d dtlog(a1+La1−θ 1)≥a1+ (1−θ)La1−θ 1 a1+La1−θ 1(a1+ 2a3). On the other hand, the inequality (4) tells us b2 3≤˜Ka1−θ 1a3 for some positive constant ˜Klarge enough with θto be ﬁxed small enough. And then d dtloga3≤a3+ 2a1+˜Ka1−θ 1, by combining with (5.2.11). Thus by choosing θ≤1 6HandL≥2˜K, d dtloga1+La1−θ 1 a3≥(a3−a1)−θLa1−θ 1 a1+La1−θ 1(a1+ 2a3)−˜Ka1−θ 1 ≥(a3−a1)−θLa1−θ 1 a1+La1−θ 1·3Ha1−˜Ka1−θ 1 ≥(a3−a1)−(3θHL+˜K)a1−θ 1 = [L−(3θHL+˜K)]a1−θ 1 ≥0 whenevera1+La1−θ 1=a3. Consequently the set {a1+La1−θ 1≥a3}is preserved. A similar argument works for the inequality in C. This completes the proof of Proposi- tion 5.2.6. The combination of the advanced maximum principle Theorem 2 .3.1 and the pinching estimates of the ODE (5.2.3) in Proposition 5.2.6 i mmediately gives the following pinching estimate for the Ricci ﬂow on a compact fo ur-manifold. Corollary 5.2.7. Suppose that the initial metric of the solution to the Ricci ﬂow on a compact four-manifold has positive curvature opera tor. Then for any ε>0 we can ﬁnd positive constant Cε<+∞such that |◦ Rm| ≤εR+Cε for allt≥0as long as the solution exists, where◦ Rmis the traceless part of the curvature operator. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 333 Proof of Theorem 5.2.2. LetMbe a compact four-manifold with positive cur- vature operator and let us evolve the metric by the Ricci ﬂow. Again the evolution equation of the scalar curvature tells us that the scalar cur vature remains positive and becomes unbounded in ﬁnite time. Pick a sequence of points xj∈Mand timesτj where the curvature at xjis as large as it has been anywhere for 0 ≤t≤τj. Dilate the metrics so that the maximum curvature |Rm(xj,τj)|= max t≤τjmax x∈M|Rm(x,t)| becomes 1 and shift the time so that the time τjbecomes the new time 0. The Klin- genberg injectivity radius estimate in Section 4.2 tells us that the injectivity radii of the rescaled metrics at the origins xjand at the new time 0 are uniformly bounded from below. Then we can apply the Hamilton’s compactness the orem (Theorem 4.1.5) to take a limit. By the pinching estimate in Corollary 5.2.7, we know that the limit metric has positive constant curvature which is either S4orRP4. Therefore the com- pact four-manifold Mis diﬀeomorphic to the sphere S4or the real projective space RP4. Remark 5.2.8. The proofs of Theorem 5.2.1 and Theorem 5.2.2 also show that the Ricci ﬂow on a compact three-manifold with positive Ricc i curvature or a com- pact four-manifold with positive curvature operator is sub sequentially converging (up to scalings) in the C∞topology to the same underlying compact manifold with a metric of positive constant curvature. Of course, this subs equential convergence is in the sense of Hamilton’s compactness theorem (Theorem 4.1 .5) which is also up to the pullbacks of diﬀeomorphisms. Actually in [58] and [59], Hamilton obtained the convergence in the stronger sense that the (rescaled) metri cs converge (in the C∞ topology) to a constant (positive) curvature metric. In the following we use the Hamilton’s strong maximum princi ple in Section 2.4 to prove Theorem 5.2.3. Proof of Theorem 5.2.3. In views of Theorem 5.2.1 and Theorem 5.2.2, we may assume the Ricci curvature (in dimension 3) and the curvatur e operator (in dimension 4) always have nontrivial kernels somewhere along the Ricci ﬂow. (i) In the case of dimension 3, we consider the evolution equa tion (1.3.5) of the Ricci curvature ∂Rab ∂t=△Rab+ 2RacbdRcd in an orthonormal frame coordinate. At each point, we diagon alizeRabwith eigen- vectorse1,e2,e3and eigenvalues λ1≤λ2≤λ3. Since R1c1dRcd=R1212R22+R1313R33 =1 2((λ3−λ2)2+λ1(λ2+λ3)), we know that if Rab≥0, thenRacbdRcd≥0. By Hamilton’s strong maximum principle (Theorem 2.2.1), there exists an interval 0 < t < δ on which the rank of Rabis constant and the null space of Rabis invariant under parallel translation and invariant in time and also lies in the null space of RacbdRcd. If the null space of Rab has rank one, then λ1= 0 andλ2=λ3>0. In this case, by De Rham decomposition theorem, the universal cover ˜Mof the compact Msplits isometrically as R×Σ2 and the curvature of Σ2has a positive lower bound. Hence Σ2is diﬀeomorphic to S2. AssumeM=R×Σ2/Γ, for some isometric subgroup Γ of R×Σ2. Note that Γ 334 H.-D. CAO AND X.-P. ZHU remains to be an isometric subgroup of R×Σ2during the Ricci ﬂow by the uniqueness (Theorem 1.2.4). Since the Ricci ﬂow on R×Σ2/Γ converges to the standard metric by Theorem 5.1.11, Γ must be an isometric subgroup of R×S2in the standard metric. If the null space of Rabhas rank greater than one, then Rab= 0 and the manifold is ﬂat. This proves Theorem 5.2.3 part (i). (ii) In the case of dimension 4, we classify the manifolds acc ording to the (re- stricted) holonomy algebra G. Note that the curvature operator has nontrivial kernel andGis the image of the the curvature operator, we see that Gis a proper subalgebra ofso(4). We divide the argument into two cases. Case1.Gis reducible. In this case the universal cover ˜Msplits isometrically as ˜M1×˜M2. By the above results on two and three dimensional Ricci ﬂow, we see that Mis diﬀeomorphic to a quotient of one of the spaces R4,R×S3,R2×S2,S2×S2by a group of ﬁxed point free isometries. As before by running the Ricci ﬂow unt il it converges and using the uniqueness (Theorem 1.2.4), we see that this group is act ually a subgroup of the isometries in the standard metrics. Case2.Gis not reducible (i.e., irreducible). If the manifold is not Einstein, then by Berger’s classiﬁcat ion theorem for holonomy groups [7], G=so(4) oru(2). Since the curvature operator is not strictly positive, G=u(2), and the universal cover ˜MofMis K¨ahler and has positive bisec- tional curvature. In this case ˜Mis biholomorphic to CP2by the result of Andreotti- Frankel [47] (also the resolution of the Frankel conjecture by Mori [96] and Siu-Yau [120]). If the manifold is Einstein, then by the block decomposition of the curvature operator matrix in four-manifolds (see the third section of Chapter 1), Rm(Λ2 +,Λ2 −) = 0. Letϕ/\e}atio\slash= 0, and ϕ=ϕ++ϕ−∈Λ2 +⊕Λ2 −, lies in the kernel of the curvature operator, then 0 =Rm(ϕ+,ϕ+) +Rm(ϕ−,ϕ−). It follows that Rm(ϕ+,ϕ+) = 0,andRm(ϕ−,ϕ−) = 0. We may assume ϕ+/\e}atio\slash= 0 (the argument for the other case is similar). We consider t he restriction of Rmto Λ2 +, since Λ2 +is an invariant subspace of Rmand the intersection of Λ2 +with the null space of Rmis nontrivial. By considering the null space of Rmand its orthogonal complement in Λ2 +, we obtain a parallel distribution of rank one in Λ2 +. This parallel distribution gives a parallel translation i nvariant two-form ω∈Λ2 +on the universal cover ˜MofM. This two-form is nondegenerate, so it induces a K¨ ahler structure of ˜M. Since the K¨ ahler metric is parallel with respect to the original metric and the manifold is irreducible, the K¨ ahler metric is proportional to the original metric. Hence the manifold ˜Mis K¨ahler-Einstein with nonnegative THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 335 curvature operator. Taking into account the irreducibilit y ofG, it follows that ˜Mis biholomorphic to CP2. Therefore the proof of Theorem 5.2.3 is completed. To end this section, we mention the generalizations of Hamil ton’s diﬀerential sphere theorem (Theorem 5.2.1 and Theorem 5.2.2) to higher d imensions. Using minimal surface theory, Micallef and Moore [88] prove d that any compact simply connected n-dimensional manifold with positive curvature operator is homeo- morphic to the sphere Sn. But it is still an open question whether a compact simply connectedn-dimensional manifold with positive curvature operator is diﬀeomorphic to the sphere Sn. It is well-known that the curvature tensor Rm={Rijkl}of a Riemannian mani- fold can be decomposed into three orthogonal components whi ch have the same sym- metries asRm: Rm=W+V+U. HereW={Wijkl}is the Weyl conformal curvature tensor, whereas V={Vijkl}and U={Uijkl}denote the traceless Ricci part and the scalar curvature par t respectively. The following pointwisely pinching sphere theorem under th e additional assumption that the manifold is compact was ﬁrst obtained by Huisken [71 ], Margerin [83], [84] and Nishikawa [101] by using the Ricci ﬂow. The compactness a ssumption was later removed by Chen and the second author in [30]. Theorem 5.2.9. Letn≥4. Suppose Mis a complete n-dimensional mani- fold with positive and bounded scalar curvature and satisﬁe s the pointwisely pinching condition |W|2+|V|2≤δn(1−ε)2|U|2, whereε>0,δ4=1 5,δ5=1 10, and δn=2 (n−2)(n+ 1),n≥6. ThenMis diﬀeomorphic to the sphere Snor a quotient of it by a ﬁnite group of ﬁxed point free isometries in the standard metric. In [30], Chen and the second author also used the Ricci ﬂow to o btain the following ﬂatness theorem for noncompact three-manifolds. Theorem 5.2.10. LetMbe a three-dimensional complete noncompact Rie- mannian manifold with bounded and nonnegative sectional cu rvature. Suppose M satisﬁes the following Ricci pinching condition Rij≥εRgij,onM, for someε>0. ThenMis ﬂat. The basic idea of proofs of these two theorems is to analyze th e asymptotic be- havior of the solution to the Ricci ﬂow. For the details, one c an consult the above cited literatures. 336 H.-D. CAO AND X.-P. ZHU 5.3. Nonsingular Solutions on Three-manifolds. We have seen in the pre- vious section that a good understanding of the long time beha viors for solutions to the Ricci ﬂow could lead to remarkable topological or geomet ric consequences for the underlying manifolds. Since one of the central themes of the Ricci ﬂow is to study the geometry and topology of three-manifolds, we will start to analyze the long time behavior of the Ricci ﬂow on a compact three-manifold by ﬁrst considering a special class of solutions (i.e., the nonsingular solutions deﬁned below) in this section. Our presentation follows closely the paper of Hamilton [65]. LetMbe a compact three-manifold. We will consider the (unnormal ized) Ricci ﬂow ∂ ∂tgij=−2Rij, and the normalized Ricci ﬂow ∂ ∂tgij=2 3rgij−2Rij wherer=r(t) is the function of the average of the scalar curvature. Reca ll that the normalized ﬂow diﬀers from the unnormalized ﬂow only by r escaling in space and time so that the total volume V=/integraltext Mdµremains constant. In this section we only consider a special class of solutions that we now deﬁne. Definition 5.3.1. Anonsingular solution of the Ricci ﬂow is one where the solution of the normalized ﬂow exists for all time 0 ≤t <∞, and the curvature remains bounded |Rm| ≤C <+∞for all time with some constant Cindependent of t. Clearly any solution to the Ricci ﬂow on a compact three-mani fold with non- negative Ricci curvature is nonsingular. Currently there a re few conditions which guarantee a solution will remain nonsingular. Nevertheles s, the ideas and arguments of this section is extremely important. One will see in Chapt er 7 that these arguments will be modiﬁed to analyze the long-time behavior of arbitra ry solutions, or even the solutions with surgery, to the Ricci ﬂow on three-manifolds . We begin with an improvement of Hamilton-Ivey pinching resu lt (Theorem 2.4.1). Theorem 5.3.2 ( Hamilton [65] ).Suppose we have a solution to the (unnormalized )Ricci ﬂow on a three–manifold which is complete with bounded curvature for each t≥0. Assume at t= 0the eigenvalues λ≥µ≥νof the curvature operator at each point are bounded below by ν≥ −1. Then at all points and all times t≥0we have the pinching estimate R≥(−ν)[log(−ν) + log(1 + t)−3] wheneverν <0. Proof. As before, we study the ODE system   dλ dt=λ2+µν, dµ dt=µ2+λν, dν dt=ν2+λµ. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 337 Consider again the function y=f(x) =x(logx−3) fore2≤x <+∞, which is increasing and convex with range −e2≤y <+∞. Its inverse function x=f−1(y) is increasing and concave on −e2≤y <+∞. For each t≥0, we consider the set K(t) of 3×3 symmetric matrices deﬁned by the inequalities: (5.3.1) λ+µ+ν≥ −3 1 +t, and (5.3.2) ν(1 +t) +f−1((λ+µ+ν)(1 +t))≥0, which is closed and convex (as we saw in the proof of Theorem 2. 4.1). By the assump- tions att= 0 and the advanced maximum principle Theorem 2.3.5, we only need to check that the set K(t) is preserved by the ODE system. SinceR=λ+µ+ν, we get from the ODE that dR dt≥2 3R2≥1 3R2 which implies that R≥ −3 1 +t,for allt≥0. Thus the ﬁrst inequality (5.3.1) is preserved. Note that the second inequality (5.3.2) is automatically satisﬁed when ( −ν)≤3/(1 +t). Now we compute from the ODE system, d dt(R (−ν)−log(−ν)) =1 (−ν)2/bracketleftbigg (−ν)·dR dt−(R+ (−ν))d(−ν) dt/bracketrightbigg =1 (−ν)2[(−ν)3+ (−ν)µ2+λ2((−ν) +µ)−λµ(ν−µ)] ≥(−ν) ≥3 (1 +t) ≥d dt[log(1 +t)−3] wheneverR= (−ν)[log(−ν) + log(1 + t)−3] and ( −ν)≥3/(1 +t). Thus the second inequality (5.3.2) is also preserved under the system of ODE . Therefore we have proved the theorem. Denote by ˆρ(t) = max {inj(x,gij(t))|x∈M} where inj(x,gij(t)) is the injectivity radius of the manifold Matxwith respect to the metricgij(t). 338 H.-D. CAO AND X.-P. ZHU Definition 5.3.3. We say a solution to the normalized Ricci ﬂow is collapsed if there is a sequence of times tk→+∞such that ˆρ(tk)→0 ask→+∞. When a nonsingular solution of the Ricci ﬂow on Mis collapsed, it follows from the work of Cheeger-Gromov [24] [25] or Cheeger-Gromov-Fukaya [26] that the manifold Mhas an F-structure and then its topology is completely understood. In the following we always assume nonsingular solutions are not collapsed. Now suppose that we have a nonsingular solution which does no t collapse. Then for arbitrary sequence of times tj→ ∞, we can ﬁnd a sequence of points xjand some δ >0 so that the injectivity radius of Matxjin the metric at time tjis at least δ. Clearly the Hamilton’s compactness theorem (Theorem 4.1. 5) also holds for the normalized Ricci ﬂow. Then by taking the xjas origins and the tjas initial times, we can extract a convergent subsequence. We call such a limit a noncollapsing limit . Of course the limit has also ﬁnite volume. However the volume of the limit may be smaller than the original one if the diameter goes to inﬁnity . The main result of this section is the following theorem of Ha milton [65]. Theorem 5.3.4 ( Hamilton [65] ).Letgij(t),0≤t <+∞, be a noncollapsing nonsingular solution of the normalized Ricci ﬂow on a compac t three-manifold M. Then either (i) there exist a sequence of times tk→+∞and a sequence of diﬀeomorphisms ϕk:M→Mso that the pull-back of the metric gij(tk)byϕkconverges in theC∞topology to a metric on Mwith constant sectional curvature; or (ii) we can ﬁnd a ﬁnite collection of complete noncompact hyp erbolic three- manifolds H1,...,Hmwith ﬁnite volume, and for all tbeyond some time T <+∞we can ﬁnd compact subsets K1,...,K mofH1,...,Hmrespectively obtained by truncating each cusp of the hyperbolic manifold s along constant mean curvature torus of small area, and diﬀeomorphisms ϕl(t),1≤l≤m, of KlintoMso that as long as tsuﬃciently large, the pull-back of the solution metricgij(t)byϕl(t)is as close as to the hyperbolic metric as we like on the compact sets K1,...,K m; and moreover if we call the exceptional part of Mthose points where they are not in the image of any ϕl, we can take the injectivity radii of the exceptional part at everywhere as s mall as we like and the boundary tori of each Klareincompressible in the sense that each ϕl injectsπ1(∂Kl)intoπ1(M). Remark 5.3.5. The exceptional part has bounded curvature and arbitrarily small injectivity radii everywhere as tlarge enough. Moreover the boundary of the exceptional part consists of a ﬁnite disjoint union of tori w ith suﬃciently small area and is convex. Then by the work of Cheeger-Gromov [24], [25] o r Cheeger-Gromov- Fukaya [26], there exists an F-structure on the exceptional part. In particular, the exceptional part is a graph manifold, which have been topolo gically classiﬁed. Hence any nonsingular solution to the normalized Ricci ﬂow is geom etrizable in the sense of Thurston (see the last section of Chapter 7 for de tails). The rest of this section is devoted to the proof of Theorem 5.3 .4. We will divide the proof into three parts. Part I: Subsequence Convergence According to Lemma 5.1.1, the scalar curvature of the normal ized ﬂow evolves by THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 339 the equation ∂ ∂tR= ∆R+ 2|Ric|2−2 3rR (5.3.3) = ∆R+ 2|◦ Ric|+2 3R(R−r) where◦ Ricis the traceless part of the Ricci tensor. As before, we denot e byRmin(t) = minx∈MR(x,t). It then follows from the maximum principle that (5.3.4)d dtRmin≥2 3Rmin(Rmin−r), which implies that if Rmin≤0 it must be nondecreasing, and if Rmin≥0 it cannot go negative again. We can then divide the noncollapsing solu tions of the normalized Ricci ﬂow into three cases. Case(1):Rmin(t)>0 for somet>0; Case(2):Rmin(t)≤0 for allt∈[0,+∞) and lim t→+∞Rmin(t) = 0; Case(3):Rmin(t)≤0 for allt∈[0,+∞) and lim t→+∞Rmin(t)<0. Let us ﬁrst consider Case (1). In this case the maximal time in terval [0,T) of the corresponding solution of the unnormalized ﬂow is ﬁnite , since the unnormalized scalar curvature ˜Rsatisﬁes ∂ ∂t˜R= ∆˜R+ 2|˜Ric|2 ≥∆˜R+2 3˜R2 which implies that the curvature of the unnormalized soluti on blows up in ﬁnite time. Without loss of generality, we may assume that for the initia l metric at t= 0, the eigenvalues ˜λ≥˜µ≥˜νof the curvature operator are bounded below by ˜ ν≥ −1. It follows from Theorem 5.3.2 that the pinching estimate ˜R≥(−˜ν)[log(−˜ν) + log(1 + t)−3] holds whenever ˜ ν <0. This shows that when the unnormalized curvature big, the negative ones are not nearly as large as the positive ones. No te that the unnormalized curvature becomes unbounded in ﬁnite time. Thus when we resc ale the unnormalized ﬂow to the normalized ﬂow, the scaling factor must go to inﬁni ty. In the nonsingular case the rescaled positive curvature stay ﬁnite, so the resc aled negative curvature (if any) go to zero. Hence we can take a noncollapsing limit for th e nonsingular solution of the normalized ﬂow so that it has nonnegative sectional cu rvature. Since the volume of the limit is ﬁnite, it follows from a resul t of Calabi and Yau [112] that the limit must be compact and the limiting manifol d is the original one. Then by the strong maximum principle as in the proof of Theore m 5.2.3 (i), either the limit is ﬂat, or it is a compact metric quotient of the product of a positively curved surface Σ2with R, or it has strictly positive curvature. By the work of Schoen -Yau [110], a ﬂat three-manifold cannot have a metric of positive scalar curvature, but our manifold does in Case (1). This rules out the possibility of a ﬂat limit. Clearly the limit is also a nonsingular solution to the normalized Ri cci ﬂow. Note that the 340 H.-D. CAO AND X.-P. ZHU curvature of the surface Σ2has a positive lower bound and is compact since it comes from the lifting of the compact limiting manifold. From Theo rem 5.1.11, we see the metric of the two-dimensional factor Σ2converges to the round two-sphere S2in the normalized Ricci ﬂow. Note also that the normalized fact ors in two-dimension and three-dimension are diﬀerent. This implies that the com pact quotient of the product Σ2×Rcannot be nonsingular, which is also ruled out for the limit. Thus the limit must have strictly positive sectional curvature. Since the convergence takes place everywhere for the compact limit, it follows that as tlarge enough the original nonsingular solution has strictly positive sectional curv ature. This in turn shows that the corresponding unnormalized ﬂow has strictly positive s ectional curvature after some ﬁnite time. Then in views of the proof of Theorem 5.2.1, i n particular the pinching estimate in Proposition 5.2.5, the limit has const ant Ricci curvature and then constant sectional curvature for three-manifolds. Th is ﬁnishes the proof in Case (1). We next consider Case (2). In this case we only need to show tha t we can take a noncollapsing limit which has nonnegative sectional curva ture. Indeed, if this is true, then as in the previous case, the limit is compact and either i t is ﬂat, or it splits as a product (or a quotient of a product) of a positively curved S2with a circle S1, or it has strictly positive curvature. But the assumption Rmin(t)≤0 for all times t≥0 in this case implies the limit must be ﬂat. Let us consider the corresponding unnormalized ﬂow ˜ gij(t) associated to the non- collapsing nonsingular solution. The pinching estimate in Theorem 5.3.2 tells us that we may assume the unnormalized ﬂow ˜ gij(t) exists for all times 0 ≤t <+∞, for otherwise, the scaling factor approaches inﬁnity as in the p revious case which implies the limit has nonnegative sectional curvature. The volume ˜V(t) of the unnormalized solution ˜gij(t) now changes. We divide the discussion into three subcases. Subcase (2.1): there is a sequence of times ˜tk→+∞such that ˜V(˜tk)→+∞; Subcase (2.2): there is a sequence of times ˜tk→+∞such that ˜V(˜tk)→0; Subcase (2.3): there exist two positive constants C1,C2such thatC1≤˜V(t)≤C2 for all 0 ≤t<+∞. For Subcase (2.1), because d˜V dt=−r˜V we have log˜V(˜tk) ˜V(0)=−/integraldisplay˜tk 0r(t)dt→+∞,ask→+∞, which implies that there exists another sequence of times, s till denoted by ˜tk, such that˜tk→+∞andr(˜tk)≤0. Lettkbe the corresponding times for the normalized ﬂow. Thus there holds for the normalized ﬂow r(tk)→0,ask→ ∞, since 0 ≥r(tk)≥Rmin(tk)→0 ask→+∞. Then /integraldisplay M(R−Rmin)dµ(tk) = (r(tk)−Rmin(tk))V→0,ask→ ∞. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 341 As we take a noncollapsing limit along the time sequence tk, we get /integraldisplay M∞Rdµ∞= 0 for the limit of the normalized solutions at the new time t= 0. ButR≥0 for the limit because lim t→+∞Rmin(t) = 0 for the nonsingular solution. So R= 0 att= 0 for the limit. Since the limit ﬂow exists for −∞< t< +∞and the scalar curvature of the limit ﬂow evolves by ∂ ∂tR= ∆R+ 2|Ric|2−2 3r∞R, t ∈(−∞,+∞) wherer∞is the limit of the function r(t) by translating the times tkas the new time t= 0. It follows from the strong maximum principle that R≡0,onM∞×(−∞,+∞). This in turn implies, in view of the above evolution equation , that Ric≡0,onM∞×(−∞,+∞). Hence this limit must be ﬂat. Since the limit M∞is complete and has ﬁnite volume, the ﬂat manifold M∞must be compact. Thus the underlying manifold M∞must agree with the original M(as a topological manifold). This says that the limit was taken onM. For Subcase (2.2), we may assume as before that for the initia l metric at t= 0 of the unnormalized ﬂow ˜ gij(t), the eigenvalues ˜λ≥˜µ≥˜νof the curvature operator satisfy ˜ν≥ −1. It then follows from Theorem 5.3.2 that ˜R≥(−˜ν)[log(−˜ν) + log(1 + t)−3],for allt≥0 whenever ˜ν <0. Lettkbe the sequence of times in the normalized ﬂow which correspo nds to the sequence of times ˜tk. Take a noncollapsing limit for the normalized ﬂow along the timestk. Since ˜V(˜tk)→0, the normalized curvatures at the times tkare reduced by multiplying the factor ( ˜V(˜tk))2 3. We claim the noncollapsing limit has nonnegative sectional curvature. Indeed if the maximum value of ( −˜ν) at the time ˜tkdoes not go to inﬁnity, the normalized eigenvalue −νat the corresponding time tkmust get rescaled to tend to zero; while if the maximum value of ( −˜ν) at the time ˜tkdoes go to inﬁnity, the maximum value of ˜Rat˜tkwill go to inﬁnity even faster from the pinching estimate, and when we normalize to keep the normalized scala r curvature Rbounded at the time tkso the normalized ( −ν) at the time tkwill go to zero. Thus in either case the noncollapsing limit has nonnegative sectional cur vature at the initial time t= 0 and then has nonnegative sectional curvature for all time st≥0. For Subcase (2.3), normalizing the ﬂow only changes quantit ies in a bounded way. As before we have the pinching estimate R≥(−ν)[log(−ν) + log(1 + t)−C] for the normalized Ricci ﬂow, where Cis a positive constant depending only on the constantsC1,C2in the assumption of Subcase (2.3). If (−ν)≤A 1 +t 342 H.-D. CAO AND X.-P. ZHU for any ﬁxed positive constant A, then ( −ν)→0 ast→+∞and we can take a noncollapsing limit which has nonnegative sectional curva ture. On the other hand if we can pick a sequence of times tk→ ∞ and points xkwhere ( −ν)(xk,tk) = max x∈M(−ν)(x,tk) satisﬁes (−ν)(xk,tk)(1 +tk)→+∞,ask→+∞, then from the pinching estimate, we have R(xk,tk) (−ν)(xk,tk)→+∞,ask→+∞. ButR(xk,tk) are uniformly bounded since normalizing the ﬂow only chang es quanti- ties in bounded way. This shows sup( −ν)(·,tk)→0 ask→+∞. Thus we can take a noncollapsing limit along tkwhich has nonnegative sectional curvature. Hence we have completed the proof of Case (2). We now come to the most interesting Case (3) where Rminincreases monotonically to a limit strictly less than zero. By scaling we can assume Rmin(t)→ −6 ast→+∞. Lemma 5.3.6. In Case (3)whereRmin→ −6ast→+∞, all noncollapsing limit are hyperbolic with constant sectional curvature −1. Proof. By (5.3.4) and the fact Rmin(t)≤ −6, we have d dtRmin(t)≥4(r(t)−Rmin(t)) and /integraldisplay∞ 0(r(t)−Rmin(t))dt<+∞. Sincer(t)−Rmin(t)≥0 andRmin(t)→ −6 ast→+∞, it follows that the function r(t) has the limit r=−6, for any convergent subsequence. And since /integraldisplay M(R−Rmin(t))dµ= (r(t)−Rmin(t))·V, it then follows that R≡ −6 for the limit . The limit still has the following evolution equation for the limiting scalar curvature ∂ ∂tR= ∆R+ 2|◦ Ric|2+2 3R(R−r). SinceR≡r≡ −6 in space and time for the limit, it follows directly that |◦ Ric| ≡0 for the limit. Thus the limit metric has λ=µ=ν=−2, so it has constant sectional curvature −1 as desired. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 343 If in the discussion above there exists a compact noncollaps ing limit, then we know that the underlying manifold Mis compact and we fall into the conclusion of Theorem 5.3.4(i) for the constant negative sectional curva ture limit. Thus it remains to show when every noncollapsing limit is a complete noncomp act hyperbolic manifold with ﬁnite volume, we have conclusion (ii) in Theorem 5.3.4. Now we ﬁrst want to ﬁnd a ﬁnite collection of persistent compl ete noncompact hyperbolic manifolds as stated in Theorem 5.3.4 (ii). Part II: Persistence of Hyperbolic Pieces We begin with the deﬁnition of the topology of C∞convergence on compact sets for mapsF:M→Nof one Riemannian manifold to another. For any compact set K⊂⊂Mand any two maps F,G:M→N, we deﬁne dK(F,G) = sup x∈Kd(F(x),G(x)) whered(y,z) is the geodesic distance from ytozonN. This gives the C0 loctopology for maps between MandN. To deﬁne Ck loctopology for any positive integer k≥1, we consider the k-jet spaceJkMof a manifold Mwhich is the collection of all (x,J1,J2,...,Jk) wherexis a point on MandJiis a tangent vector for 1 ≤i≤kdeﬁned by the ithcovariant derivative Ji=∇i ∂ ∂tγ(0) for a path γpassing through the point xwith γ(0) =x. A smooth map F:M→Ninduces a map JkF:JkM→JkN deﬁned by JkF(x,J1,...,Jk) = (F(x),∇∂ ∂t(F(γ))(0),...,∇k ∂ ∂t(F(γ))(0)) whereγis a path passing through the point xwithJi=∇i ∂ ∂tγ(0),1≤i≤k. Deﬁne the k-jet distance betweenFandGon a compact set K⊂⊂Mby dCk(K)(F,G) =dBJkK(JkF,JkG) whereBJkKconsists of all k-jets (x,J1,...,Jk) withx∈Kand |J1|2+|J2|2+···+|Jk|2≤1. Then the convergence in the metric dCk(K)for all positive integers kand all compact setsKdeﬁnes the topology of C∞convergence on compact sets for the space of maps. We will need the following Mostow type rigidity result. Lemma 5.3.7. For any complete noncompact hyperbolic three-manifold Hwith ﬁnite volume with metric h, we can ﬁnd a compact set KofHsuch that for every integerkand everyε >0, there exist an integer qand aδ >0with the following property: if Fis a diﬀeomorphism of Kinto another complete noncompact hyperbolic three-manifold ˜Hwith no fewer cusps (thanH),ﬁnite volume with metric ˜hsuch that /ba∇dblF∗˜h−h/ba∇dblCq(K)<δ 344 H.-D. CAO AND X.-P. ZHU then there exists an isometry IofHto˜Hsuch that dCk(K)(F,I)<ε. Proof. First we claim that His isometric to ˜Hfor an appropriate choice of compact set K, positive integer qand positive number δ. Letl:H → Rbe a function deﬁned at each point by the length of the shortest non-contra ctible loop starting and ending at this point. Denote the Margulis constant by µ. Then by Margulis lemma (see for example [55] or [76]), for any 0 < ε0<1 2µ, the setl−1([0,ε0])⊂ Hconsists of ﬁnitely many components and each of these components is is ometric to a cusp or to a tube. Topologically, a tube is just a solid torus. Let ε0be even smaller than one half of the minimum of the lengths of the all closed geodes ics on the tubes. Then l−1([0,ε0]) consists of ﬁnite number of cusps. Set K0=l−1([ε0,∞)). The boundary ofK0consists of ﬂat tori with constant mean curvatures. Note tha t each embedded torus in a complete hyperbolic three-manifold with ﬁnite vo lume either bounds a solid torus or is isotopic to a standard torus in a cusp. The diﬀeomo rphismFimplies the boundaryF(∂K0) are embedded tori. If one of components bounds a solid torus , then asδsuﬃciently small and qsuﬃciently large, ˜Hwould have fewer cusps than H, which contradicts with our assumption. Consequently, ˜His diﬀeomorphic to F(o K0). Here o K0is the interior of the set K0. Since His diﬀeomorphic too K0,His diﬀeomorphic to˜H. Hence by Mostow’s rigidity theorem (see [97] and [107]), His isometric to ˜H. So we can assume ˜H=H. ForK=K0, we argue by contradiction. Suppose there is somek >0 andε >0 so that there exist sequences of integers qj→ ∞,δj→0+ and diﬀeomorphisms Fjmapping KintoHwith /ba∇dblF∗ jh−h/ba∇dblCqj(K)<δj and dCk(K)(Fj,I)≥ε for all isometries IofHto itself. We can extract a subsequence of Fjconvergent to a mapF∞withF∗ ∞h=honK. We need to check that F∞is still a diﬀeomorphism on K. SinceF∞is a local diﬀeomorphism and is the limit of diﬀeomorphisms, we can ﬁnd an inverse of F∞on F∞(o K). SoF∞is a diﬀeomorphism ono K. We claim the image of the boundary can not touch the image of the interior. Indeed, if F∞(x1) =F∞(x2) withx1∈∂Kand x2∈o K, then we can ﬁnd x3∈o Knearx1andx4∈o Knearx2withF∞(x3) =F∞(x4), sinceF∞is a local diﬀeomorphism. This contradicts with the fact tha tF∞is a diﬀeomorphism ono K. This proves our claim. Hence, the only possible overlap is a t the boundary. But the image F∞(∂K) is strictly concave, this prevents the boundary from touching itself. We conclude that the mapping F∞is a diﬀeomorphism on K, hence an isometry. To extendF∞to a global isometry, we argue as follows. For each truncated cusp end of K, the area of constant mean curvature ﬂat torus is strictly de creasing. Since F∞takes each such torus to another of the same area, we see that F∞takes the foliation of an end by constant mean curvature ﬂat tori to ano ther such foliation. So F∞takes cusps to cusps and preserves their foliations. Note th at the isometric type THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 345 of a cusp is just the isometric type of the torus, more precise ly, let (N,dr2+e−2rgV) be a cusp (where gVis the ﬂat metric on the torus V), 0<a<b are two constants, any isometry of N∩l−1[a,b] to itself is just an isometry of V. Hence the isometry F∞ can be extended to the whole cusps. This gives a global isomet ryIcontradicting our assumption when jlarge enough. The proof of the Lemma 5.3.7 is completed. In order to obtain the persistent hyperbolic pieces stated i n Theorem 5.3.4 (ii), we will need to use a special parametrization given by harmon ic maps. Lemma 5.3.8. Let(X, g)be a compact Riemannian manifold with strictly negative Ricci curvature and with strictly concave boundar y. Then there are positive integerl0and small number ε0>0such that for each positive integer l≥l0and positive number ε≤ε0we can ﬁnd positive integer qand positive number δ >0 such that for every metric ˜gonXwith||˜g−g||Cq(X)≤δwe can ﬁnd a unique diﬀeomorphism FofXto itself so that (a)F: (X,g)→(X,˜g)is harmonic, (b)Ftakes the boundary ∂Xto itself and satisﬁes the free boundary condi- tionthat the normal derivative ∇NFofFat the boundary is normal to the boundary, (c)dCl(X)(F,Id)<ε, whereIdis the identity map. Proof. Let Φ(X,∂X ) be the space of maps of Xto itself which take ∂Xto itself. Then Φ(X,∂X ) is a Banach manifold and the tangent space to Φ( X,∂X ) at the identity is the space of vector ﬁelds V=Vi∂ ∂xitangent to the boundary. Consider the map sending F∈Φ(X,∂X ) to the pair {∆F,(∇NF)//}consisting of the harmonic map Laplacian and the tangential component (in the target) o f the normal derivative ofFat the boundary. By using the inverse function theorem, we on ly need to check that the derivative of this map is an isomorphism at the ident ity with ˜g=g. Let{xi}i=1,...,nbe a local coordinates of ( X,g) and {yα}α=1,...,nbe a local co- ordinates of ( X,˜g). The harmonic map Laplacian of F: (X,g)→(X,˜g) is given in local coordinates by (∆F)α= ∆(Fα) +gij(˜Γα βγ◦F)∂Fβ ∂xi∂Fγ ∂xj where ∆(Fα) is the Laplacian of the function FαonXand˜Γα βγis the connection of ˜g. LetFbe a one-parameter family with F|s=0=IdanddF ds|s=0=V, a smooth vector ﬁeld on Xtangent to the boundary (with respect to g). At an arbitrary given pointx∈X, we choose the coordinates {xi}i=1,...,nso that Γi jk(x) = 0.We compute at the point xwith ˜g=g, d ds/vextendsingle/vextendsingle/vextendsingle s=0(∆F)α= ∆(Vα) +gij/parenleftbigg∂ ∂xkΓα ij/parenrightbigg Vk. Since (∇iV)α=∇iVα+ (Γα iβ◦F)Vβ, we have, at s= 0 and the point x, (∆V)α= ∆(Vα) +gij∂ ∂xiΓα jkVk. 346 H.-D. CAO AND X.-P. ZHU Thus we obtain d ds|s=0(∆F)α= (∆V)α+gij/parenleftbigg∂ ∂xkΓα ij−∂ ∂xiΓα jk/parenrightbigg Vk(5.3.5) = (∆V)α+gαiRikVk. Since (∇NF)(F−1(x)) =Ni(F−1(x))∂Fj ∂xi(F−1(x))∂ ∂xj(x) on∂X, we have d ds|s=0{(∇NF)//}=d ds|s=0(∇NF− /a\}b∇acketle{t∇ NF,N/a\}b∇acket∇i}htN) (5.3.6) =d ds|s=0(∇NF)−/angbracketleftbiggd ds|s=0∇NF,N/angbracketrightbigg N − /a\}b∇acketle{t∇ NF,∇VN/a\}b∇acket∇i}htN|s=0− /a\}b∇acketle{t∇ NF,N/a\}b∇acket∇i}ht∇VN|s=0 =/parenleftbiggd ds|s=0∇NF/parenrightbigg //− ∇ VN =/parenleftbigg −V(Ni)∂ ∂xi+N(Vj)∂ ∂xj/parenrightbigg //−II(V) = [N,V]//−II(V) = (∇NV)//−2II(V) whereIIis the second fundamental form of the boundary (as an automor phism of T(∂X)). Thus by (5.3.5) and (5.3.6), the kernel of the map sending F∈Φ(X,∂X ) to the pair {∆F,(∇NF)//}is the space of solutions of elliptic boundary value problem (5.3.7)  ∆V+ Ric (V) = 0 on X V⊥= 0, at∂X, (∇NV)//−2II(V) = 0,at∂X, whereV⊥is the normal component of V. Now using these equations and integrating by parts gives /integraldisplay /integraldisplay X|∇V|2=/integraldisplay /integraldisplay XRic(V,V) + 2/integraldisplay ∂XII(V,V). SinceRc < 0 andII <0 we conclude that the kernel is trivial. Clearly this ellipt ic boundary value is self-adjoint because of the free boundary condition. Thus the cokernel is trivial also. This proves the lemma. Now we can prove the persistence of hyperbolic pieces. Let gij(t),0≤t<+∞, be a noncollapsing nonsingular solution of the normalized R icci ﬂow on a compact three-manifold M. Assume that any noncollapsing limit of the nonsingular sol ution is a complete noncompact hyperbolic three-manifold with ﬁnit e volume. Consider all the possible hyperbolic limits of the given nonsingular soluti on, and among them choose one such complete noncompact hyperbolic three-manifold Hwith the least possible number of cusps. In particular, we can ﬁnd a sequence of times tk→+∞and a THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 347 sequence of points PkonMsuch that the marked three-manifolds ( M,g ij(tk),Pk) converge in the C∞ loctopology to Hwith hyperbolic metric hijand marked point P∈ H. For any small enough a>0 we can truncate each cusp of Halong a constant mean curvature torus of area awhich is uniquely determined; the remainder we denote by Ha. Clearly as a→0 theHaexhaust H. Pick a suﬃciently small number a >0 to truncate cusps so that Lemma 5.3.7 is applicable for the comp act set K=Ha. Choose an integer l0large enough and an ε0suﬃciently small to guarantee from Lemma 5.3.8 the uniqueness of the identity map Idamong maps close to Idas a harmonic mapFfromHato itself with taking ∂Hato itself, with the normal derivative of Fat the boundary of the domain normal to the boundary of the tar get, and with dCl0(Ha)(F,Id)< ε0. Then choose positive integer q0and small number δ0>0 from Lemma 5.3.7 such that if ˜Fis a diﬀeomorphism of Hainto another complete noncompact hyperbolic three-manifold ˜Hwith no fewer cusps (than H), ﬁnite volume with metric ˜hijsatisfying ||˜F∗˜hij−hij||Cq0(Ha)≤δ0, then there exists an isometry IofHto˜Hsuch that (5.3.8) dCl0(Ha)(˜F,I)<ε0. And we further require q0andδ0from Lemma 5.3.8 to guarantee the existence of harmonic diﬀeomorphism from ( Ha,˜gij) to (Ha,hij) for any metric ˜ gijonHawith ||˜gij−hij||Cq0(Ha)≤δ0. By deﬁnition, there exist a sequence of exhausting compact s etsUkofH(each Uk⊃ H a) and a sequence of diﬀeomorphisms FkfromUkintoMsuch thatFk(P) =Pk and||F∗ kgij(tk)−hij||Cm(Uk)→0 ask→+∞for all positive integers m. Note that ∂Hais strictly concave and we can foliate a neighborhood of ∂Hawith constant mean curvature hypersurfaces where the area ahas a nonzero gradient. As the approximat- ing mapsFk: (Uk,hij)→(M,g ij(tk)) are close enough to isometries on this collar of ∂Ha, the metrics gij(tk) onMwill also admit a unique constant mean curvature hy- persurface with the same area anearFk(∂Ha)(⊂M) by the inverse function theorem. Thus we can change the map Fkby an amount which goes to zero as k→ ∞ so that nowFk(∂Ha) has constant mean curvature with the area a. Furthermore, by applying Lemma 5.3.8 we can again change Fkby an amount which goes to zero as k→ ∞ so as to makeFka harmonic diﬀeomorphism and take ∂Hato the constant mean curvature hypersurface Fk(∂Ha) and also satisfy the free boundary condition that the norma l derivative of Fkat the boundary of the domain is normal to the boundary of the target. Hence for arbitrarily given positive integer q≥q0and positive number δ<δ 0, there exists a positive integer k0such that for the modiﬁed harmonic diﬀeomorphism Fk, whenk≥k0, ||F∗ kgij(tk)−hij||Cq(Ha)<δ. For each ﬁxed k≥k0, by the implicit function theorem we can ﬁrst ﬁnd a constant mean curvature hypersurface near Fk(∂Ha) inMwith the metric gij(t) fortclose to tkand with the same area for each component since ∂Hais strictly concave and a neighborhood of ∂Hais foliated by constant mean curvature hypersurfaces where the areaahas a nonzero gradient and Fk: (Ha,hij)→(M,g ij(tk)) is close enough to an isometry and gij(t) varies smoothly. Then by applying Lemma 5.3.8 we can smooth ly 348 H.-D. CAO AND X.-P. ZHU continue the harmonic diﬀeomorphism Fkforward in time a little to a family of harmonic diﬀeomorphisms Fk(t) from HaintoMwith the metric gij(t), withFk(tk) = Fk, where each Fk(t) takes∂Hainto the constant mean curvature hypersurface we just found in ( M,g ij(t)) and satisﬁes the free boundary condition, and also satisﬁ es ||F∗ k(t)gij(t)−hij||Cq(Ha)<δ. We claim that for all suﬃciently large k, we can smoothly extend the harmonic dif- feomorphism Fkto the family harmonic diﬀeomorphisms Fk(t) with ||F∗ k(t)gij(t)− hij||Cq(Ha)≤δon a maximal time interval tk≤t≤ωk(ortk≤t < ω kwhen ωk= +∞); and ifωk<+∞, then (5.3.9) ||F∗ k(ωk)gij(ωk)−hij||Cq(Ha)=δ. Clearly the above argument shows that the set of twhere we can extend the harmonic diﬀeomorphisms as desired is open. To verify claim (5.3.9), we thus only need to show that if we have a family of harmonic diﬀeomorphis msFk(t) such as we desire for tk≤t < ω (<+∞), we can take the limit of Fk(t) ast→ωto get a harmonic diﬀeomorphism Fk(ω) satisfying ||F∗ k(ω)gij(ω)−hij||Cq(Ha)≤δ, and if ||F∗ k(ω)gij(ω)−hij||Cq(Ha)<δ, then we can extend Fk(ω) forward in time a little (i.e., we can ﬁnd a constant mean curvature hypersurface near Fk(ω)(∂Ha) inMwith the metric gij(t) for eachtclose toωand with the same area afor each component). Note that (5.3.10) ||F∗ k(t)gij(t)−hij||Cq(Ha)<δ. fortk≤t<ω and the metrics gij(t) fortk≤t≤ωare uniformly equivalent. We can ﬁnd a subsequence tn→ωfor whichFk(tn) converge to Fk(ω) inCq−1(Ha) and the limit map has ||F∗ k(ω)gij(ω)−hij||Cq−1(Ha)≤δ. We need to check that Fk(ω) is still a diﬀeomorphism. We at least know Fk(ω) is a local diﬀeomorphism, and Fk(ω) is the limit of diﬀeomorphisms, so the only possibility of overlap is at the boundary. Hence we use the fact that Fk(ω)(∂Ha) is still strictly concave since qis large and δis small to prevent the boundary from touching itself. ThusFk(ω) is a diﬀeomorphism. A limit of harmonic maps is harmonic, so Fk(ω) is a harmonic diﬀeomorphism from HaintoMwith the metric gij(ω). Moreover Fk(ω) takes∂Hato the constant mean curvature hypersurface ∂(Fk(ω)(Ha)) of the area ain (M,g ij(ω)) and continue to satisfy the free boundary condition. As a c onsequence of the standard regularity result of elliptic partial diﬀeren tial equations (see for example [48]), the map Fk(ω)∈C∞(Ha) and then from (5.3.10) we have ||F∗ k(ω)gij(ω)−hij||Cq(Ha)≤δ. If||F∗ k(ω)gij(ω)−hij||Cq(Ha)=δ, we then ﬁnish the proof of the claim. So we may assume that ||F∗ k(ω)gij(ω)−hij||Cq(Ha)< δ. We want to show that Fk(ω) can be extended forward in time a little. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 349 We argue by contradiction. Suppose not, then we consider the new sequence of the manifolds Mwith metric gij(ω) and the origins Fk(ω)(P). SinceFk(ω) are close to isometries, the injectivity radii of the metrics gij(ω) atFk(ω)(P) do not go to zero, and we can extract a subsequence which converges to a hyperbo lic limit/tildewideHwith the metric/tildewidehijand the origin /tildewidePand with ﬁnite volume. The new limit /tildewideHhas at least as many cusps as the old limit H, since we choose Hwith cusps as few as possible. By the deﬁnition of convergence, we can ﬁnd a sequence of compac t sets/tildewideBkexhausting /tildewideHand containing /tildewideP, and a sequence of diﬀeomorphisms /tildewideFkof neighborhoods of /tildewideBk intoMwith/tildewideFk(/tildewideP) =Fk(ω)(P) such that for each compact set /tildewideBin/tildewideHand each integerm ||/tildewideF∗ k(gij(ω))−/tildewidehij||Cm( /CTB)→0 ask→+∞. For large enough kthe set/tildewideFk(/tildewideBk) will contain all points out to any ﬁxed distance we need from the point Fk(ω)(P), and then /tildewideFk(/tildewideBk)⊃Fk(ω)(Ha) since the points of Hahave a bounded distance from PandFk(ω) are reasonably close to preserving the metrics. Hence we can form the compos ition Gk=/tildewideF−1 k◦Fk(ω) :Ha→/tildewideH. Arbitrarily ﬁx δ′∈(δ,δ0). Since the/tildewideFkare as close to preserving the metric as we like, we have ||G∗ k/tildewidehij−hij||Cq(Ha)<δ′ for all suﬃciently large k. By Lemma 5.3.7, we deduce that there exists an isometry IofHto/tildewideH, and then ( M,g ij(ω),Fk(ω)(P)) (on compact subsets) is very close to (H,hij,P) as long as δsmall enough and klarge enough. Since Fk(ω)(∂Ha) is strictly concave and the foliation of a neighborhood of Fk(ω)(∂Ha) by constant mean curva- ture hypersurfaces has the area as a function with nonzero gr adient, by the implicit function theorem, there exists a unique constant mean curva ture hypersurface with the same area anearFk(ω)(∂Ha) inMwith the metric gij(t) fortclose toω. Hence, whenksuﬃciently large, Fk(ω) can be extended forward in time a little. This is a contradiction and we have proved claim (5.3.9). We further claim that there must be some ksuch thatωk= +∞(i.e., we can smoothly continue the family of harmonic diﬀeomorphisms Fk(t) for alltk≤t<+∞, in other words, there must be at least one hyperbolic piece pe rsisting). We argue by contradiction. Suppose for each klarge enough, we can continue the family Fk(t) for tk≤t≤ωk<+∞with ||F∗ k(ωk)gij(ωk)−hij||Cq(Ha)=δ. Then as before, we consider the new sequence of the manifolds Mwith metrics gij(ωk) and origins Fk(ωk)(P). For suﬃciently large k, we can obtain diﬀeomorphisms /tildewideFkof neighborhoods of /tildewideBkintoMwith/tildewideFk(/tildewideP) =Fk(ωk)(P) which are as close to preserving the metric as we like, where /tildewideBkis a sequence of compact sets, exhausting some hyperbolic three-manifold ˜H, of ﬁnite volume and with no fewer cusps (than H), and 350 H.-D. CAO AND X.-P. ZHU containing ˜P; moreover, the set /tildewideFk(/tildewideBk) will contain all the points out to any ﬁxed distance we need from the point Fk(ωk)(P); and hence /tildewideFk(/tildewideBk)⊇Fk(ωk)(Ha) sinceHais at bounded distance from PandFk(ωk) is reasonably close to preserving the metrics. Then we can form the composition Gk=/tildewideF−1 k◦Fk(ωk) :Ha→˜H. Since the/tildewideFkare as close to preserving the metric as we like, for any /tildewideδ>δ we have ||G∗ k/tildewidehij−hij||Cq(Ha)</tildewideδ for large enough k. Then a subsequence of Gkconverges at least in Cq−1(Ha) topology to a mapG∞ofHainto/tildewideH. By the same reason as in the argument of previous two paragraphs, the limit map G∞is a smooth harmonic diﬀeomorphism from Hainto /tildewideHwith the metric ˜hij, and takes ∂Hato a constant mean curvature hypersurface G∞(∂Ha) of ( ˜H,/tildewidehij) with the area a, and also satisﬁes the free boundary condition. Moreover we still have (5.3.11) ||G∗ ∞/tildewidehij−hij||Cq(Ha)=δ. Now by Lemma 5.3.7 we deduce that there exists an isometry IofHto/tildewideHwith dCl0(Ha)(G∞,I)<ε0. By usingIto identify/tildewideHaandHa, we see that the map I−1◦G∞is a harmonic diﬀeomorphism of Hato itself which satisﬁes the free boundary condition and dCl0(Ha)(I−1◦G∞,Id)<ε0. From the uniqueness in Lemma 5.3.8 we conclude that I−1◦G∞=Idwhich contra- dicts with (5.3.11). This shows at least one hyperbolic piec e persists. Moreover the pull-back of the solution metric gij(t) byFk(t), fortk≤t <+∞, is as close to the hyperbolic metric hijas we like. We can continue to form other persistent hyperbolic pieces i n the same way as long as there are any points Pkoutside of the chosen pieces where the injectivity radius at times tk→ ∞ are all at least some ﬁxed positive number ρ>0. The only modiﬁcation in the proof is to take the new limit Hto have the least possible number of cusps out of all remaining possible limits. Note that the volume of the normalized Ricci ﬂow is constant i n time. Therefore by combining with Margulis lemma (see for example [55] [76]) , we have proved that there exists a ﬁnite collection of complete noncompact hype rbolic three-manifolds H1,...,Hmwith ﬁnite volume, a small number a>0 and a time T <+∞such that for alltbeyondTwe can ﬁnd diﬀeomorphisms ϕl(t) of (Hl)aintoM, 1≤l≤m, so that the pull-back of the solution metric gij(t) byϕl(t) is as close to the hyperbolic metrics as we like and the exceptional part of Mwhere the points are not in the image of anyϕlhas the injectivity radii everywhere as small as we like. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 351 Part III: Incompressibility We remain to show that the boundary tori of any persistent hyp erbolic piece are incompressible, in the sense that the fundamental group of t he torus injects into that of the whole manifold. The argument of this part is a paraboli c version of Schoen and Yau’s minimal surface argument in [109, 110, 111]. LetBbe a small positive number and assume the above positive numb erais much smaller than B. Denote by Maa persistent hyperbolic piece of the manifold M truncated by boundary tori of area awith constant mean curvature and denote by Mc a=M\◦ Mathe part of Mexterior to Ma. Thus there is a persistent hyperbolic pieceMB⊂Maof the manifold Mtruncated by boundary tori of area Bwith constant mean curvature. We also denote by Mc B=M\◦ MB. By Van Kampen’s Theorem, if π1(∂MB) injects into π1(Mc B) then it injects into π1(M) also. Thus we only need to showπ1(∂MB) injects into π1(Mc B). We will argue by contradiction. Let Tbe a torus in ∂MB. Supposeπ1(T) does not inject intoπ1(Mc B), then by Dehn’s Lemma the kernel is a cyclic subgroup of π1(T) generated by a primitive element. The work of Meeks-Yau [86] or Meeks-Simon-Yau [87] shows that among all disks in Mc Bwhose boundary curve lies in Tand generates the kernel, there is a smooth embedded disk normal to the boun dary which has the least possible area. Let A=A(t) be the area of this disk. This is deﬁned for all t suﬃciently large. We will show that A(t) decreases at a rate bounded away from zero which will be a contradiction. Let us compute the rate at which A(t) changes under the Ricci ﬂow. We need to showA(t) decrease at least at a certain rate, and since A(t) is the minimum area to bound any disk in the given homotopy class, it suﬃces to ﬁnd so me such disk whose area decreases at least that fast. We choose this disk as foll ows. Pick the minimal disk at timet0, and extend it smoothly a little past the boundary torus sinc e the minimal disk is normal to the boundary. For times ta little bigger than t0, the boundary torus may need to move a little to stay constant mean curvature with areaBas the metrics change, but we leave the surface alone and take the bounding d isk to be the one cut oﬀ from it by the new torus. The change of the area ˜A(t) of such disk comes from the change in the metric and the change in the boundary. For the change in the metric, we choose an orthonormal frame X,Y,Z at a point xin the disk so that XandYare tangent to the disk while Zis normal and compute the rate of change of the area element dσon the disk as ∂ ∂tdσ=1 2(gij)T/parenleftbigg2 3r(gij/parenrightbiggT −2(Rij)T)dσ =/bracketleftbigg2 3r−Ric (X,X)−Ric (Y,Y)/bracketrightbigg dσ, since the metric evolves by the normalized Ricci ﬂow. Here ( ·)Tdenotes the tangential projections on the disk. Notice the torus Tmay move in time to preserve constant mean curvature and constant area B. Suppose the boundary of the disk evolves with a normal velocity N. The change of the area at boundary along a piece of length ds is given by Nds. Thus the total change of the area ˜A(t) is given by d˜A dt=/integraldisplay /integraldisplay/parenleftbigg2 3r−Ric (X,X)−Ric (Y,Y)/parenrightbigg dσ+/integraldisplay ∂Nds. 352 H.-D. CAO AND X.-P. ZHU Note that Ric (X,X) + Ric (Y,Y) =R(X,Y,X,Y ) +R(X,Z,X,Z ) +R(Y,X,Y,X ) +R(Y,Z,Y,Z ) =1 2R+R(X,Y,X,Y ). By the Gauss equation, the Gauss curvature Kof the disk is given by K=R(X,Y,X,Y ) + detII whereIIis the second fundamental form of the disk in Mc B. This gives at t=t0, dA dt≤/integraldisplay /integraldisplay/parenleftbigg2 3r−1 2R/parenrightbigg dσ−/integraldisplay /integraldisplay (K−detII)dσ+/integraldisplay ∂Nds Since the bounding disk is a minimal surface, we have detII≤0. The Gauss-Bonnet Theorem tells us that for a disk /integraldisplay /integraldisplay Kdσ+/integraldisplay ∂kds= 2π wherekis the geodesic curvature of the boundary. Thus we obtain (5.3.12)dA dt≤/integraldisplay /integraldisplay/parenleftbigg2 3r−1 2R/parenrightbigg dσ+/integraldisplay ∂kds+/integraldisplay ∂Nds−2π. Recall that we are assuming Rmin(t) increases monotonically to −6 ast→+∞. By the evolution equation of the scalar curvature, d dtRmin(t)≥4(r(t)−Rmin(t)) and then /integraldisplay∞ 0(r(t)−Rmin(t))dt<+∞. This implies that r(t)→ −6 ast→+∞by using the derivatives estimate for the curvatures. Thus for every ε>0 we have 2 3r−1 2R≤ −(1−ε) fortsuﬃciently large. And then the ﬁrst term on RHS of (5.3.12) is bounded above by /integraldisplay /integraldisplay/parenleftbigg2 3r−1 2R/parenrightbigg dσ≤ −(1−ε)A. The geodesic curvature kof the boundary of the minimal disk is the acceleration of a curve moving with unit speed along the intersection of the d isk with the torus; THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 353 since the disk and torus are normal, this is the same as the sec ond fundamental form of the torus in the direction of the curve of intersection. No w if the metric were actually hyperbolic, the second fundamental form of the tor us would be exactly 1 in all directions. Note that the persistent hyperbolic pieces are as close to the standard hyperbolic as we like. This makes that the second term of RHS o f (5.3.12) is bounded above by /integraldisplay ∂kds≤(1 +ε0)L for some suﬃciently small positive number ε0>0, whereLis the length of the boundary curve. Also since the metric on the persistent hype rbolic pieces are close to the standard hyperbolic as we like, its change under the no rmalized Ricci ﬂow is as small as we like; So the motion of the constant mean curvatu re torus of ﬁxed area Bwill have a normal velocity Nas small as we like. This again makes the third term of RHS of (5.3.12) bounded above by /integraldisplay ∂Nds≤ε0L. Combining these estimates, we obtain (5.3.13)dA dt≤(1 + 2ε0)L−(1−ε0)A−2π on the persistent hyperbolic piece, where ε0is some suﬃciently small positive number. We next need to bound the length Lin terms of the area A. Sinceais much smaller thanB, for largetthe metric is as close as we like to the standard hyperbolic on e; not just on the persistent hyperbolic piece MBbut as far beyond as we like. Thus for a long distance into Mc Bthe metric will look nearly like a standard hyperbolic cuspl ike collar. Let us ﬁrst recall a special coordinate system on the standar d hyperbolic cusp projecting beyond torus T1in∂H1as follows. The universal cover of the ﬂat torus T1can be mapped conformally to the x-yplane so that the deck transformation of T1 become translations in xandy, and so that the Euclidean area of the quotient is 1; then these coordinates are unique up to a translation. The hy perbolic cusp projecting beyond the torus T1in∂H1can be parametrized by {(x,y,z )∈R3|z>0}with the hyperbolic metric (5.3.14) ds2=dx2+dy2+dz2 z2. Note that we can make the solution metric, in an arbitrarily l arge neighborhood of the torusT(of∂MB), as close to hyperbolic as we wish (in the sense that there ex ists a diﬀeomorphism from a large neighborhood of the torus TB(of∂HB) on the standard hyperbolic cusp to the above neighborhood of the torus T(of∂MB) such that the pull-back of the solution metric by the diﬀeomorphism is as c lose to the hyperbolic metric as we wish). Then by using this diﬀeomorphism (up to a s light modiﬁcation) we can parametrize the cusplike tube of Mc Bprojecting beyond the torus Tin∂MB by{(x,y,z )|z≥ζ}where the height ζis chosen so that the torus in the hyperbolic cusp at height ζhas the area B. Now consider our minimal disk, and let L(z) be the length of the curve of the intersection of the disk with the torus at height zin the above coordinate system, 354 H.-D. CAO AND X.-P. ZHU and also let A(z) be the area of the part of the disk between ζandz. We now want to derive a monotonicity formula on the area A(z) for the minimal surface. For almost every zthe intersection of the disk with the torus at height zis a smooth embedded curve or a ﬁnite union of them by the standard transversality theorem. If there is more than one curve, at least one of them i s not homotopic to a point inTand represents the primitive generator in the kernel of π1(T) such that a part of the original disk beyond height zcontinues to a disk that bounds it. We extend this disk back to the initial height ζby dropping the curve straight down. Let ˜L(z) be the length of the curve we picked at height z; of course ˜L(z)≤L(z) with equality if it is the only piece. Let ˜L(w) denote the length of the same curve in the x-yplane dropped down to height wforζ≤w≤z. In the hyperbolic space we would have ˜L(w) =z w˜L(z) exactly. In our case there is a small error proportional to ˜L(z) and we can also take it proportional to the distance z−wby which it drops since ˜L(w)|w=z=˜L(z) and the solution metric is close to the hyperbolic in the C∞ loctopology. Thus, for arbitrarily givenδ >0 andζ∗> ζ, as the solution metric is suﬃciently close to hyperbolic, w e have |˜L(w)−z w˜L(z)| ≤δ(z−w)˜L(z) for allzandwinζ≤w≤z≤ζ∗. Now given εandζ∗pickδ= 2ε/ζ∗. Then (5.3.15) ˜L(w)≤z w˜L(z)/bracketleftbigg 1 +2ε(z−w) w/bracketrightbigg . When we drop the curve vertically for the construction of the new disk we get an area ˜A(z) betweenζandzgiven by ˜A(z) = (1 +o(1))/integraldisplayz ζ˜L(w) wdw. Here and in the following o(1) denotes various small error quantities as the solution metric close to hyperbolic. On the other hand if we do not drop vertically we pick up even more area, so the area A(z) of the original disk between ζandzhas (5.3.16) A(z)≥(1−o(1))/integraldisplayz ζL(w) wdw. Since the original disk minimized among all disks bounded a c urve in the primitive generator of the kernel of π1(T), and the new disk beyond the height zis part of the original disk, we have A(z)≤˜A(z) and then by combining with (5.3.15), /integraldisplayz ζL(w) wdw≤(1 +o(1))z˜L(z)/integraldisplayz ζ/bracketleftbigg1−2ε w2+2εz w3/bracketrightbigg dw ≤(1 +o(1))L(z)(z−ζ) ζ/bracketleftbigg 1 +ε/parenleftbiggz−ζ ζ/parenrightbigg/bracketrightbigg . THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 355 Here we used the fact that ˜L(z)≤L(z). Since the solution metric is suﬃciently close to hyperbolic, we have d dz/parenleftbigg/integraldisplayz ζL(w) wdw/parenrightbigg =L(z) z ≥(1−o(1))ζ z(z−ζ)/bracketleftbigg 1−ε/parenleftbiggz−ζ ζ/parenrightbigg/bracketrightbigg/integraldisplayz ζL(w) wdw ≥/bracketleftbigg1 z−ζ−1 + 2ε z/bracketrightbigg/integraldisplayz ζL(w) wdw, or equivalently (5.3.17)d dzlog/braceleftbiggz1+2ε (z−ζ)/integraldisplayz ζL(w) wdw/bracerightbigg ≥0. This is the desired monotonicity formula for the area A(z). It follows directly from (5.3.16) and (5.3.17) that z1+2ε (z−ζ)A(z)≥(1−o(1))ζ2εL(ζ), or equivalently L(ζ)≤(1 +o(1))/parenleftbiggz ζ/parenrightbigg2εz z−ζA(z) for allz∈[ζ,ζ∗]. Since the solution metric, in an arbitrarily large neighb orhood of the torusT(of∂MB), as close to hyperbolic as we wish, we may assume that ζ∗is so large that√ζ∗>ζand√ζ∗√ζ∗−ζis close to 1, and also ε>0 is so small that (√ζ∗ ζ)2εis close to 1. Thus for arbitrarily small δ0>0, we have (5.3.18) L(ζ)≤(1 +δ0)A/parenleftig/radicalbig ζ∗/parenrightig . Now recall that (5.3.13) states dA dt≤(1 + 2ε0)L−(1−ε0)A−2π. We now claim that if (1 + 2ε0)L−(1−ε0)A≥0 thenL=L(ζ) is uniformly bounded from above. Indeed by the assumption we have A(ζ∗)≤(1 + 2ε0) (1−ε0)L(ζ) sinceA(ζ∗)≤A. By combining with (5.3.16) we have some z0∈(√ζ∗,ζ∗) satisfying L(z0) z0/parenleftig ζ∗−/radicalbig ζ∗/parenrightig ≤/integraldisplayζ∗ √ζ∗L(w) wdw ≤(1 +o(1))A(ζ∗) ≤(1 +o(1))/parenleftbigg1 + 2ε0 1−ε0/parenrightbigg L(ζ). 356 H.-D. CAO AND X.-P. ZHU Thus forζ∗suitably large, by noting that the solution metric on a large neighborhood ofT(of∂MB) is suﬃciently close to hyperbolic, we have (5.3.19) L(z0)≤(1 + 4ε0)z0 ζ∗L(ζ) for somez0∈(√ζ∗,ζ∗). It is clear that we may assume the intersection curve betwe en the minimal disk with the torus at this height z0is smooth and embedded. If the intersection curve at the height z0has more than one piece, as before one of them will represent the primitive generator in the kernel of π1(T), and we can ignore the others. Let us move (the piece of) the intersection curve on the torus at heightz0through as small as possible area in the same homotopy class of π1(T) to a curve which is a geodesic circle in the ﬂat torus coming from our special coor dinates, and then drop this geodesic circle vertically in the special coordinates to obtain another new disk. We will compare the area of this new disk with the original min imal disk as follows. Denote byGthe length of the geodesic circle in the standard hyperbolic cusp at height 1. Then the length of the geodesic circle at height z0will beG/z0. Observe that given an embedded curve of length lcircling the cylinder S1×Rof circumference wonce, it is possible to deform the curve through an area not bi gger thanlwinto a meridian circle. Note that (the piece of) the intersection c urve represents the primitive generator in the kernel of π1(T). Note also that the solution metric is suﬃciently close to the hyperbolic metric. Then the area of the deformation fr om (the piece of) the intersection curve on the torus at height z0to the geodesic circle at height z0is bounded by (1 +o(1))/parenleftbiggG z0/parenrightbigg ·L(z0). The area to drop the geodesic circle from height z0to heightζis bounded by (1 +o(1))/integraldisplayz0 ζG w2dw. Hence comparing the area of the original minimal disk to that of this new disk gives A(z0)≤(1 +o(1))G/bracketleftbiggL(z0) z0+/parenleftbigg1 ζ−1 z0/parenrightbigg/bracketrightbigg . By (5.3.18), (5.3.19) and the fact that z0∈(√ζ∗,ζ∗), this in turn gives L(ζ)≤(1 +δ0)A(z0) ≤(1 +δ0)G/bracketleftbigg (1 + 4ε0)L(ζ) ζ∗+1 ζ/bracketrightbigg . Sinceζ∗is suitably large, we obtain L(ζ)≤2G/ζ This gives the desired assertion since Gis ﬁxed from the geometry of the limit hyper- bolic manifold Handζis very large as long as the area Bof∂MBsmall enough. Thus the combination of (5.3.13), (5.3.18) and the assertio n implies that either d dtA≤ −2π, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 357 or d dtA≤(1 + 2ε0)L−(1−ε0)A−2π ≤(1 + 2ε0)2G ζ−2π ≤ −π, since the solution metric on a very large neighborhood of the torusT(of∂MB) is suﬃciently close to hyperbolic and ζis very large as the area Bof∂MBsmall enough. This is impossible because A≥0 and the persistent hyperbolic pieces go on forever. The contradiction shows that π1(T) in fact injects into π1(Mc B). This proves that π1(∂MB) injects into π1(M). Therefore we have completed the proof of Theorem 5.3.4. 6. Ancient κ-solutions. Let us consider a solution of the Ricci ﬂow on a com- pact manifold. If the solution blows up in ﬁnite time (i.e., t he maximal solution exists only on a ﬁnite time interval), then as we saw in Chapter 4 a seq uence of rescalings of the solution around the singularities converge to a solut ion which exists at least on the time interval ( −∞,T) for some ﬁnite number T. Furthermore, by Perelman’s no local collapsing theorem I (Theorem 3.3.2), we see that th e limit isκ-noncollapsed on all scales for some positive constant κ. In addition, if the dimension n= 3 then the Hamilton-Ivey pinching estimate implies that the limit ing solution must have nonnegative curvature operator. We call a solution to the Ricci ﬂow an ancientκ-solution if it is complete (either compact or noncompact) and deﬁned on an ancient time interva l (−∞,T) withT >0, has nonnegative curvature operator and bounded curvature, and isκ-noncollapsed on all scales for some positive constant κ. In this chapter we study ancient κ-solutions of the Ricci ﬂow. We will obtain crucial curvature estimates of such solutions and determin e their structures in lower dimensional cases. 6.1. Preliminaries. We ﬁrst present a useful geometric property, given by Chen and the second author in [34], for complete noncompact Riema nnian manifolds with nonnegative sectional curvature. Let (M,g ij) be ann-dimensional complete Riemannian manifold and let εbe a positive constant. We call an open subset N⊂Manε-neck of radius rif (N,r−2gij) isε-close, in the C[ε−1]topology, to a standard neck Sn−1×I, where Sn−1 is the round ( n−1)-sphere with scalar curvature 1 and Iis an interval of length 2 ε−1. The following result is, to some extent, in similar spirit of Yau’s volume lower bound estimate [128]. Proposition 6.1.1 ( Chen-Zhu [34] ).There exists a positive constant ε0=ε0(n) such that every complete noncompact n-dimensional Riemannian manifold (M,g ij) of nonnegative sectional curvature has a positive constant r0such that any ε-neck of radiusron(M,g ij)withε≤ε0must haver≥r0. Proof. We argue by contradiction. Suppose there exist a sequence of positive constantsεα→0 and a sequence of n-dimensional complete noncompact Riemannian manifolds (Mα,gα ij) such that for each ﬁxed α, there exists a sequence of εα-necksNk of radius at most 1 /kinMαwith centers Pkdivergent to inﬁnity. 358 H.-D. CAO AND X.-P. ZHU Fix a point Pon the manifold Mαand connect each PktoPby a minimizing geodesicγk. By passing to a subsequence we may assume the angle θklbetween geodesicγkandγlatPis very small and tends to zero as k,l→+∞, and the length ofγk+1is much bigger than the length of γk. Let us connect PktoPlby a minimizing geodesicηkl. For each ﬁxed l>k, let˜Pkbe a point on the geodesic γlsuch that the geodesic segment from Pto˜Pkhas the same length as γkand consider the triangle ∆PPk˜PkinMαwith vertices P,Pkand˜Pk. By comparing with the corresponding triangle in the Euclidean plane R2whose sides have the same corresponding lengths, Toponogov’s comparison theorem implies d(Pk,˜Pk)≤2 sin/parenleftbigg1 2θkl/parenrightbigg ·d(Pk,P). Sinceθklis very small, the distance from Pkto the geodesic γlcan be realized by a geodesicζklwhich connects Pkto a pointP′ kon the interior of the geodesic γland has length at most 2 sin(1 2θkl)·d(Pk,P).Clearly the angle between ζklandγlat the intersection point P′ kisπ 2. Consider αto be ﬁxed and suﬃciently large. We claim that for large enough k, each minimizing geodesic γlwithl>k, connecting PtoPl, goes through the neck Nk. Suppose not; then the angle between γkandζklatPkis close to either zero or π sincePkis in the center of an εα-neck andαis suﬃciently large. If the angle between γkandζklatPkis close to zero, we consider the triangle ∆ PPkP′ kinMαwith vertices P,Pk, andP′ k. Note that the length between PkandP′ kis much smaller than the lengths from PkorP′ ktoP. By comparing the angles of this triangle with those of the corresponding triangle in the Euclidean plane with the same corresponding lengths and using Toponogov’s comparison theorem, we ﬁnd that it is i mpossible. Thus the angle between γkandζklatPkis close toπ. We now consider the triangle ∆ PkP′ kPl inMαwith the three sides ζkl,ηkland the geodesic segment from P′ ktoPlonγl. We have seen that the angle of ∆ PkP′ kPlatPkis close to zero and the angle at P′ kisπ 2. By comparing with corresponding triangle ¯∆¯Pk¯P′ k¯Plin the Euclidean plane R2whose sides have the same corresponding lengths, Toponogov’s com parison theorem implies ∠¯Pl¯Pk¯P′ k+∠¯Pl¯P′ k¯Pk≤∠PlPkP′ k+∠PlP′ kPk<3 4π. This is impossible since the length between ¯Pkand¯P′ kis much smaller than the length from ¯Plto either ¯Pkor¯P′ k. So we have proved each γlwithl>k passes through the neckNk. Hence by taking a limit, we get a geodesic ray γemanating from Pwhich passes through all the necks Nk,k= 1,2,...,except a ﬁnite number of them. Throwing these ﬁnite number of necks away, we may assume γpasses through all necks Nk, k= 1,2,....Denote the center sphere of NkbySk, and their intersection points with γbypk∈Sk∩γ,k= 1,2,.... Take a sequence of points γ(m) withm= 1,2,.... For each ﬁxed neck Nk, arbitrarily choose a point qk∈Nknear the center sphere Skand draw a geodesic segmentγkmfromqktoγ(m). Now we claim that for any neck Nlwithl > k,γkm will pass through Nlfor all suﬃciently large m. We argue by contradiction. Let us place all the necks Nihorizontally so that the geodesicγpasses through each Nifrom the left to the right. We observe that the geodesic segment γkmmust pass through the right half of Nk; otherwise γkmcannot be minimal. Then for large enough m, the distance from plto the geodesic segment THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 359 γkmmust be achieved by the distance from plto some interior point pk′ofγkm. Let us draw a minimal geodesic ηfromplto the interior point pk′with the angle at the intersection point pk′∈η∩γkmto beπ 2.Suppose the claim is false. Then the angle betweenηandγatplis close to 0 or πsinceεαis small. If the angle between ηandγatplis close to 0, we consider the triangle ∆ plpk′γ(m) and construct a comparison triangle ¯∆¯pl¯pk′¯γ(m) in the plane with the same corre- sponding length. Then by Toponogov’s comparison theorem, w e see the sum of the inner angles of the comparison triangle ¯∆¯pl¯pk′¯γ(m) is less than 3 π/4, which is impos- sible. If the angle between ηandγatplis close toπ, by drawing a minimal geodesic ξfromqktopl, we see that ξmust pass through the right half of Nkand the left half ofNl; otherwise ξcannot be minimal. Thus the three inner angles of the tri- angle ∆plpk′qkare almost 0 ,π/2, and 0 respectively. This is also impossible by the Toponogov comparison theorem. Hence we have proved that the geodesic segment γkmpasses through Nlform large enough. Consider the triangle ∆ pkqkγ(m) with two long sides pkγ(m)(⊂γ) andqkγ(m)(= γkm). For anys>0, choose points ˜ pkonpkγ(m) and ˜qkonqkγ(m) withd(pk,˜pk) = d(qk,˜qk) =s. By Toponogov’s comparison theorem, we have/AId(˜pk,˜qk) d(pk, qk) /AJ2 =d(˜pk, γ(m))2+d(˜qk, γ(m))2−2d(˜pk, γ(m))d(˜qk, γ(m))cos ¯∡(˜pkγ(m)˜qk) d(pk, γ(m))2+d(qk, γ(m))2−2d(pk, γ(m))d(qk, γ(m))cos ¯∡(pkγ(m)qk) ≥d(˜pk, γ(m))2+d(˜qk, γ(m))2−2d(˜pk, γ(m))d(˜qk, γ(m))cos ¯∡(˜pkγ(m)˜qk) d(pk, γ(m))2+d(qk, γ(m))2−2d(pk, γ(m))d(qk, γ(m))cos ¯∡(˜pkγ(m)˜qk) =(d(˜pk, γ(m))−d(˜qk, γ(m)))2+ 2d(˜pk, γ(m))d(˜qk, γ(m))(1−cos¯∡(˜pkγ(m)˜qk)) (d(˜pk, γ(m))−d(˜qk, γ(m)))2+ 2d(pk, γ(m))d(qk, γ(m))(1−cos¯∡(˜pkγ(m)˜qk)) ≥d(˜pk, γ(m))d(˜qk, γ(m)) d(pk, γ(m))d(qk, γ(m)) →1 asm→ ∞, where ¯∡(pkγ(m)qk) and ¯∡(˜pkγ(m)˜qk) are the corresponding angles of the comparison triangles. Lettingm→ ∞, we see that γkmhas a convergent subsequence whose limit γk is a geodesic ray passing through all Nlwithl>k. Let us denote by pj=γ(tj),j= 1,2,.... From the above computation, we deduce that d(pk,qk)≤d(γ(tk+s),γk(s)) for alls>0. Letϕ(x) = lim t→+∞(t−d(x,γ(t))) be the Busemann function constructed from the rayγ. Note that the level set ϕ−1(ϕ(pj))∩Njis close to the center sphere Sjfor anyj= 1,2,.... Now letqkbe any ﬁxed point in ϕ−1(ϕ(pk))∩Nk. By the deﬁnition of Busemann function ϕassociated to the ray γ, we see that ϕ(γk(s1))−ϕ(γk(s2)) = s1−s2for anys1,s2≥0. Consequently, for each l >k, by choosing s=tl−tk, we seeγk(tl−tk)∈ϕ−1(ϕ(pl))∩Nl.Sinceγ(tk+tl−tk) =pl, it follows that d(pk,qk)≤d(pl,γk(s)). 360 H.-D. CAO AND X.-P. ZHU withs=tl−tk>0. This implies that the diameter of ϕ−1(ϕ(pk))∩Nkis not greater than the diameter of ϕ−1(ϕ(pl))∩Nlfor anyl > k, which is a contradiction for l much larger than k. Therefore we have proved the proposition. In [63], Hamilton discovered an important repulsion princi ple (cf. Theorem 21.4 of [63]) about the inﬂuence of a bump of strictly positive cur vature in a complete noncompact manifold of nonnegative sectional curvature. N amely minimal geodesic paths that go past the bump have to avoid it. As a consequence h e obtained a ﬁnite bump theorem (cf. Theorem 21.5 of [63]) that gives a bound on t he number of bumps of curvature. LetMbe a complete noncompact Riemannian manifold with nonnegat ive sec- tional curvature K≥0. A geodesic ball B(p,r) of radiusrcentered at a point p∈M is called a curvature β-bump if sectional curvature K≥β/r2at all points in the ball. The ball B(p,r) is calledλ-remote from an origin Oifd(p,O)≥λr. Finite Bump Theorem (Hamilton [63]) .For everyβ >0there exists λ<∞ such that in any complete manifold of nonnegative sectional curvature there are at most a ﬁnite number of disjoint balls which are λ-remote curvature β-bumps. This ﬁnite bump theorem played an important role in Hamilton ’s study of the behavior of singularity models at inﬁnity and in the dimensi on reduction argument he developed for the Ricci ﬂow (cf. Section 22 of [63], see also [ 29] for application to the K¨ ahler-Ricci ﬂow and uniformization problem in complex di mension two). A special consequence of the ﬁnite bump theorem is that if we have a comp lete noncompact solution to the Ricci ﬂow on an ancient time interval −∞< t < T withT >0 satisfying certain local injectivity radius bound, with cu rvature bounded at each time and with asymptotic scalar curvature ratio A= limsupRs2=∞, then we can ﬁnd a sequence of points pjgoing to ∞(as in the following Lemma 6.1.3) such that a cover of the limit of dilations around these points at time t= 0 splits as a product with a ﬂat factor. The following result, obtained by Chen and the second author in [34], is in similar spirit as Hamilton’s ﬁnite bumps theorem and its consequence. The advantage is that we will get in the limit of dilations a produ ct of the line with a lower dimensional manifold, instead of a quotient of such a p roduct. Proposition 6.1.2 ( Chen-Zhu [34] ).Suppose (M,g ij)is a complete n- dimensional Riemannian manifold with nonnegative section al curvature. Let P∈M be ﬁxed, and Pk∈Ma sequence of points and λka sequence of positive numbers withd(P,Pk)→+∞andλkd(P,Pk)→+∞. Suppose also that the marked manifolds (M,λ2 kgij,Pk)converge in the C∞ loctopology to a Riemannian manifold /tildewiderM. Then the limit/tildewiderMsplits isometrically as the metric product of the form R×N, whereNis a Riemannian manifold with nonnegative sectional curvature . Proof. Let us denote by |OQ|=d(O,Q) the distance between two points O,Q∈ M. Without loss of generality, we may assume that for each k, (6.1.1) 1 + 2 |PPk| ≤ |PPk+1|. Draw a minimal geodesic γkfromPtoPkand a minimal geodesic σkfromPkto Pk+1, both parametrized by arclength. We may further assume (6.1.2) θk=|∡(˙γk(0),˙γk+1(0))|<1 k. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 361 By assumption, the sequence ( M,λ2 kgij,Pk) converges (in the C∞ loctopology) to a Riemannian manifold ( /tildewiderM,/tildewidegij,/tildewideP) with nonnegative sectional curvature. By a further choice of subsequences, we may also assume γkandσkconverge to geodesic rays /tildewideγ and/tildewideσstarting at/tildewidePrespectively. We will prove that ˜ γ∪˜σforms a line in /tildewiderM, and then by the Toponogov splitting theorem [89] the limit /tildewiderMmust be splitted as R×N. We argue by contradiction. Suppose /tildewideγ∪/tildewideσis not a line; then for each k, there exist two points Ak∈γkandBk∈σksuch that as k→+∞, (6.1.3)  λkd(Pk,Ak)→A>0, λkd(Pk,Bk)→B >0, λkd(Ak,Bk)→C >0, butA+B >C. ((((((((((((((((((((((((((((((((((( ((((((( PPk Pk+1 AkδkBkσk γk Now draw a minimal geodesic δkfromAktoBk. Consider comparison triangles ¯△¯Pk¯P¯Pk+1and¯△¯Pk¯Ak¯BkinR2with |¯Pk¯P|=|PkP|,|¯Pk¯Pk+1|=|PkPk+1|,|¯P¯Pk+1|=|PPk+1|, and|¯Pk¯Ak|=|PkAk|,|¯Pk¯Bk|=|PkBk|,|¯Ak¯Bk|=|AkBk|. By Toponogov’s comparison theorem [8], we have (6.1.4) ∡¯Ak¯Pk¯Bk≥∡¯P¯Pk¯Pk+1. On the other hand, by (6.1.2) and using Toponogov’s comparis on theorem again, we have (6.1.5) ∡¯Pk¯P¯Pk+1≤∡PkPPk+1<1 k, and since |¯Pk¯Pk+1|>|¯P¯Pk|by (6.1.1), we further have (6.1.6) ∡¯Pk¯Pk+1¯P≤∡¯Pk¯P¯Pk+1<1 k. Thus the above inequalities (6.1.4)-(6.1.6) imply that ∡¯Ak¯Pk¯Bk>π−2 k. 362 H.-D. CAO AND X.-P. ZHU Hence (6.1.7) |¯Ak¯Bk|2≥ |¯Ak¯Pk|2+|¯Pk¯Bk|2−2|¯Ak¯Pk| · |¯Pk¯Bk|cos/parenleftbigg π−2 k/parenrightbigg . Multiplying the above inequality by λ2 kand letting k→+∞, we get C≥A+B which contradicts (6.1.3). Therefore we have proved the proposition. LetMbe ann-dimensional complete noncompact Riemannian manifold. Pi ck an originO∈M. Letsbe the geodesic distance to the origin OofM, andRthe scalar curvature. Recall that in Chapter 4 we have deﬁned the asymptotic scalar curvature ratio A= limsup s→+∞Rs2. We now state a useful lemma of Hamilton (Lemma 22.2 in [63]) ab out picking local (almost) maximum curvature points at inﬁnity. Lemma 6.1.3. Given a complete noncompact Riemannian manifold with bound ed curvature and with asymptotic scalar curvature ratio A= limsup s→+∞Rs2= +∞, we can ﬁnd a sequence of points xjdivergent to inﬁnity, a sequence of radii rjand a sequence of positive numbers δj→0such that (i)R(x)≤(1 +δj)R(xj)for allxin the ballB(xj,rj)of radiusrjaroundxj, (ii)r2 jR(xj)→+∞, (iii)λj=d(xj,O)/rj→+∞, (iv) the balls B(xj,rj)are disjoint, whered(xj,O)is the distance of xjfrom the origin O. Proof. Pick a sequence of positive numbers ǫj→0, then choose Aj→+∞so thatAjǫ2 j→+∞. Letσjbe the largest number such that sup{R(x)d(x,O)2|d(x,O)≤σj} ≤Aj. Then there exists some yj∈Msuch that R(yj)d(yj,O)2=Ajandd(yj,O) =σj. Now pickxj∈Mso thatd(xj,O)≥σjand R(xj)≥1 1 +ǫjsup{R(x)|d(x,O)≥σj}. Finally pick rj=ǫjσj. We check the properties (i)-(iv) as follows. (i) Ifx∈B(xj,rj)∩ {d(·,O)≥σj}, we have R(x)≤(1 +ǫj)R(xj) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 363 by the choice of the point xj; while ifx∈B(xj,rj)∩ {d(·,O)≤σj}, we have R(x)≤Aj/d(x,O)2 ≤1 (1−ǫj)2(Aj/σ2 j) =1 (1−ǫj)2R(yj) ≤(1 +ǫj) (1−ǫj)2R(xj), sinced(x,O)≥d(xj,O)−d(x,xj)≥σj−rj= (1−ǫj)σj. Thus we have obtained R(x)≤(1 +δj)R(xj),∀x∈B(xj,rj), whereδj=(1+ǫj) (1−ǫj)2−1→0 asj→+∞. (ii) By the choices of rj,xjandyj, we have r2 jR(xj) =ǫ2 jσ2 jR(xj) ≥ǫ2 jσ2 j/bracketleftbigg1 1 +ǫjR(yj)/bracketrightbigg =ǫ2 j 1 +ǫjAj→+∞,asj→+∞. (iii) Since d(xj,O)≥σj=rj/ǫj, it follows that λj=d(xj,O)/rj→+∞as j→+∞. (iv) For any x∈B(xj,rj), the distance from the origin d(x,O)≥d(xj,O)−d(x,xj) ≥σj−rj = (1−ǫj)σj→+∞,asj→+∞. Thus any ﬁxed compact set does not meet the balls B(xj,rj) for large enough j. If we pass to a subsequence, the balls will all avoid each other. The above point picking lemma of Hamilton, as written down in Lemma 22.2 of [63], requires the curvature of the manifold to be bounded. W hen the manifold has unbounded curvature, we will appeal to the following simple lemma. Lemma 6.1.4. Given a complete noncompact Riemannian manifold with un- bounded curvature, we can ﬁnd a sequence of points xjdivergent to inﬁnity such that for each positive integer j, we have |Rm(xj)| ≥j, and |Rm(x)| ≤4|Rm(xj)| forx∈B(xj,j√ |Rm(xj)|). Proof. Eachxjcan be constructed as a limit of a ﬁnite sequence {yi}, deﬁned as follows. Let y0be any ﬁxed point with |Rm(y0)| ≥j. Inductively, if yicannot be 364 H.-D. CAO AND X.-P. ZHU taken asxj, then there is a yi+1such that   |Rm(yi+1)|>4|Rm(yi)|, d(yi,yi+1)/lessorequalslantj/radicalbig |Rm(yi)|. Thus we have |Rm(yi)|>4i|Rm(y0)| ≥4ij, d(yi,y0)≤ji/summationdisplay k=11/radicalbig 4k−1j<2/radicalbig j. Since the manifold is smooth, the sequence {yi}must be ﬁnite. The last element ﬁts. 6.2. Asymptotic Shrinking Solitons. We begin with the study of the asymp- totic behavior of an ancient κ-solutiongij(x,t), onM×(−∞,T) withT >0, to the Ricci ﬂow as t→ −∞ . Pick an arbitrary point ( p,t0)∈M×(−∞,0] and recall from Chapter 3 that τ=t0−t,fort<t 0, l(q,τ) =1 2√τinf/braceleftbigg/integraldisplayτ 0√s/parenleftig R(γ(s),t0−s) +|˙γ(s)|2 gij(t0−s)/parenrightig ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleγ: [0,τ]→Mwith γ(0) =p, γ(τ) =q/bracerightbigg and ˜V(τ) =/integraldisplay M(4πτ)−n 2exp(−l(q,τ))dVt0−τ(q). We ﬁrst observe that Corollary 3.2.6 also holds for the gener al complete manifold M. Indeed, since the scalar curvature is nonnegative, the fun ction ¯L(·,τ) = 4τl(·,τ) achieves its minimum on Mfor each ﬁxed τ >0. Thus the same argument in the proof of Corollary 3.2.6 shows there exists q=q(τ) such that (6.2.1) l(q(τ),τ)≤n 2 for eachτ >0. Recall from (3.2.11)-(3.2.13), the Li-Yau-Perelman dista ncelsatisﬁes the follow- ing ∂ ∂τl=−l τ+R+1 2τ3/2K, (6.2.2) |∇l|2=−R+l τ−1 τ3/2K, (6.2.3) ∆l≤ −R+n 2τ−1 2τ3/2K, (6.2.4) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 365 and the equality in (6.2.4) holds everywhere if and only if we are on a gradient shrinking soliton. Here K=/integraltextτ 0s3/2Q(X)ds, Q (X) is the trace Li-Yau-Hamilton quadratic given by Q(X) =−Rτ−R τ−2/a\}b∇acketle{t∇R,X/a\}b∇acket∇i}ht+ 2Ric(X,X) andXis the tangential (velocity) vector ﬁeld of an L-shortest curve γ: [0,τ]→M connecting ptoq. By applying the trace Li-Yau-Hamilton inequality (Corolla ry 2.5.5) to the ancient κ-solution, we have Q(X) =−Rτ−R τ−2/a\}b∇acketle{t∇R,X/a\}b∇acket∇i}ht+ 2Ric(X,X) ≥ −R τ and hence K=/integraldisplayτ 0s3/2Q(X)ds ≥ −/integraldisplayτ 0√sRds ≥ −L(q,τ). Thus by (6.2.3) we get (6.2.5) |∇l|2+R≤3l τ. We now state and prove a result of Perelman [103] about the asy mptotic shapes of ancientκ-solutions as the time t→ −∞ . Theorem 6.2.1 ( Perelman [103] ).Letgij(·,t),−∞< t < T with someT >0, be a nonﬂat ancient κ-solution for some κ>0. Then there exist a sequence of points qkand a sequence of times tk→ −∞ such that the scalings of gij(·,t)aroundqkwith factor |tk|−1and with the times tkshifting to the new time zero converge to a nonﬂat gradient shrinking soliton in C∞ loctopology. Proof. Clearly, we may assume that the nonﬂat ancient κ-solution is not a gradient shrinking soliton. For the arbitrarily ﬁxed ( p,t0), letq(τ)(τ=t0−t) be chosen as in (6.2.1) with l(q(τ),τ)≤n 2. We only need to show that the scalings of gij(·,t) around q(τ) with factor τ−1converge along a subsequence of τ→+∞to a nonﬂat gradient shrinking soliton in the C∞ loctopology. We ﬁrst claim that for any A≥1, one can ﬁnd B=B(A)<+∞such that for every ¯τ >1 there holds (6.2.6) l(q,τ)≤BandτR(q,t0−τ)≤B, whenever1 2¯τ≤τ≤A¯τandd2 t0−¯τ 2(q,q(¯τ 2))≤A¯τ. 366 H.-D. CAO AND X.-P. ZHU Indeed, by using (6.2.5) at τ=¯τ 2, we have /radicalbigg l(q,¯τ 2)≤/radicalbiggn 2+ sup{|∇√ l|} ·dt0−¯τ 2/parenleftig q,q/parenleftig¯τ 2/parenrightig/parenrightig (6.2.7) ≤/radicalbiggn 2+/radicalbigg 3 2¯τ·√ A¯τ =/radicalbiggn 2+/radicalbigg 3A 2, and R/parenleftig q,t0−¯τ 2/parenrightig ≤3l(q,¯τ 2) (¯τ 2)(6.2.8) ≤6 ¯τ/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2 , forq∈Bt0−¯τ 2(q(¯τ 2),√ A¯τ). Recall that the Li-Yau-Hamilton inequality implies that the scalar curvature of the ancient solution is pointwise no ndecreasing in time. Thus we know from (6.2.8) that (6.2.9) τR(q,t0−τ)≤6A/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2 whenever1 2¯τ≤τ≤A¯τandd2 t0−¯τ 2(q,q(¯τ 2))≤A¯τ. By (6.2.2) and (6.2.3) we have ∂l ∂τ+1 2|∇l|2=−l 2τ+R 2. This together with (6.2.9) implies that ∂l ∂τ≤ −l 2τ+3A τ/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2 i.e., ∂ ∂τ(√τl)≤3A√τ/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2 whenever1 2¯τ≤τ≤A¯τandd2 t0−¯τ 2/parenleftbig q,q/parenleftbig¯τ 2/parenrightbig/parenrightbig ≤A¯τ.Hence by integrating this diﬀer- ential inequality, we obtain √τl(q,τ)−/radicalbigg ¯τ 2l/parenleftig q,¯τ 2/parenrightig ≤6A/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2√τ and then by (6.2.7), l(q,τ)≤l/parenleftig q,¯τ 2/parenrightig + 6A/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2 (6.2.10) ≤7A/parenleftigg/radicalbiggn 2+/radicalbigg 3A 2/parenrightigg2 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 367 whenever1 2¯τ≤τ≤A¯τandd2 t0−¯τ 2(q,q(¯τ 2))≤A¯τ.So we have proved claim (6.2.6). Recall that gij(τ) =gij(·,t0−τ) satisﬁes (gij)τ= 2Rij. Let us take the scaling of the ancient κ-solution around q(¯τ 2) with factor (¯τ 2)−1, i.e., ˜gij(s) =2 ¯τgij/parenleftig ·,t0−s¯τ 2/parenrightig wheres∈[0,+∞). Claim (6.2.6) says that for all s∈[1,2A] and allqsuch that dist2 ˜gij(1)(q,q(¯τ 2))≤A,we have ˜R(q,s) =¯τ 2R(q,t0−s¯τ 2)≤B. Now taking into account the κ-noncollapsing assumption and Theorem 4.2.2, we can use Ham ilton’s compactness theorem (Theorem 4.1.5) to obtain a sequence ¯ τk→+∞such that the marked evolving manifolds ( M,˜g(k) ij(s),q(¯τk 2)),with ˜g(k) ij(s) =2 ¯τkgij(·,t0−s¯τk 2) and s∈[1,+∞), converge to a manifold ( ¯M,¯gij(s),¯q) withs∈[1,+∞), where ¯gij(s) is also a solution to the Ricci ﬂow on ¯M. Denote by ˜lkthe corresponding Li-Yau-Perelman distance of ˜ g(k) ij(s). It is easy to see that ˜lk(q,s) =l(q,¯τk 2s),fors∈[1,+∞).From (6.2.5), we also have (6.2.11) |∇˜lk|2 ˜g(k) ij+˜R(k)≤6˜lk, where ˜R(k)is the scalar curvature of the metric ˜ g(k) ij. Claim (6.2.6) says that ˜lkare uniformly bounded on compact subsets of M×[1,+∞) (with the corresponding origins q(¯τk 2)). Thus the above gradient estimate (6.2.11) implies that t he functions ˜lktend (up to a subsequence) to a function ¯lwhich is a locally Lipschitz function on ¯M. We know from (6.2.2)-(6.2.4) that the Li-Yau-Perelman dist ance˜lksatisﬁes the following inequalities: (6.2.12) ( ˜lk)s−∆˜lk+|∇˜lk|2−˜R(k)+n 2s≥0, (6.2.13) 2∆ ˜lk− |∇˜lk|2+˜R(k)+˜lk−n s≤0. We next show that the limit ¯lalso satisﬁes the above two inequalities in the sense of distributions. Indeed the above two inequalities can be rew ritten as (6.2.14)/parenleftbigg∂ ∂s− △+/tildewideR(k)/parenrightbigg/parenleftig (4πs)−n 2exp(−/tildewidelk)/parenrightig ≤0, (6.2.15) −(4△ −/tildewideR(k))e− /CTlk 2+˜lk−n se− /CTlk 2≤0, in the sense of distributions. Note that the estimate (6.2.1 1) implies that ˜lk→¯lin theC0,α locnorm for any 0 < α < 1.Thus the inequalities (6.2.14) and (6.2.15) imply that the limit lsatisﬁes (6.2.16)/parenleftbigg∂ ∂s− △+R/parenrightbigg/parenleftbig (4πs)−n 2exp(−l)/parenrightbig ≤0, (6.2.17) −(4△ −R)e−l 2+¯l−n se−l 2≤0, 368 H.-D. CAO AND X.-P. ZHU in the sense of distributions. Denote by ˜V(k)(s) Perelman’s reduced volume of the scaled metric ˜ g(k) ij(s). Since ˜lk(q,s) =l(q,¯τk 2s), we see that ˜V(k)(s) =˜V(¯τk 2s) where ˜Vis Perelman’s reduced volume of the ancient κ-solution. The monotonicity of Perelman’s reduced volume (Theorem 3.2.8) then implies that (6.2.18) lim k→∞˜V(k)(s) =¯V,fors∈[1,2], for some nonnegative constant ¯V. (We remark that by the Jacobian comparison theorem (Theorem 3.2.7), (3.2.18) and (3.2.19), the integrand of ˜V(k)(s) is bounded by (4πs)−n 2exp(−˜lk(X,s))˜J(k)(s)≤(4π)−n 2exp(−|X|2) onTpM, where ˜J(k)(s) is the L-Jacobian of the L-exponential map of the metric ˜g(k) ij(s) atTpM. Thus we can apply the dominant convergence theorem to get th e convergence in (6.2.18). But we are not sure whether the limi ting¯Vis exactly Perel- man’s reduced volume of the limiting manifold ( ¯M,¯gij(s)), because the points q(¯τk 2) may diverge to inﬁnity. Nevertheless, we can ensure that ¯Vis not less than Perelman’s reduced volume of the limit.) Note by (6.2.5) that ˜V(k)(2)−˜V(k)(1) (6.2.19) =/integraldisplay2 1d ds(˜V(k)(s))ds =/integraldisplay2 1ds/integraldisplay M/parenleftbigg∂ ∂s−∆ +˜R(k)/parenrightbigg/parenleftig (4πs)−n 2exp(−˜lk)/parenrightig dV˜g(k) ij(s). Thus we deduce that in the sense of distributions, (6.2.20)/parenleftbigg∂ ∂s−∆ +¯R/parenrightbigg/parenleftbig (4πs)−n 2exp(−¯l)/parenrightbig = 0, and (4∆−¯R)e−¯l 2=¯l−n se−¯l 2 or equivalently, (6.2.21) 2∆ ¯l− |∇¯l|2+¯R+¯l−n s= 0, on¯M×[1,2]. Thus by applying standard parabolic equation theory to ( 6.2.20) we ﬁnd that ¯lis actually smooth. Here we used (6.2.2)-(6.2.4) to show tha t the equality in (6.2.16) implies the equality in (6.2.17). Set v= [s(2∆¯l− |∇¯l|2+¯R) +¯l−n]·(4πs)−n 2e−¯l. Then by applying Lemma 2.6.1, we have (6.2.22)/parenleftbigg∂ ∂s−∆ +¯R/parenrightbigg v=−2s|¯Rij+∇i∇j¯l−1 2s¯gij|2·(4πs)−n 2e−¯l. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 369 We see from (6.2.21) that the LHS of the equation (6.2.22) is i dentically zero. Thus the limit metric ¯ gijsatisﬁes (6.2.23) ¯Rij+∇i∇j¯l−1 2s¯gij= 0, so we have shown the limit is a gradient shrinking soliton. To show that the limiting gradient shrinking soliton is nonﬂ at, we ﬁrst show that the constant function ¯V(s) is strictly less than 1. Consider Perelman’s reduced volum e ˜V(τ) of the ancient κ-solution. By using Perelman’s Jacobian comparison theore m (Theorem 3.2.7), (3.2.18) and (3.2.19) as before, we have ˜V(τ) =/integraldisplay (4πτ)−n 2e−l(X,τ)J(τ)dX ≤/integraldisplay TpM(4π)−n 2e−|X|2dX = 1. Recall that we have assumed the nonﬂat ancient κ-solution is not a gradient shrinking soliton. Thus for τ >0,we must have ˜V(τ)<1. Since the limiting function ¯V(s) is the limit of ˜V(¯τk 2s) with ¯τk→+∞, we deduce that the constant ¯V(s) is strictly less than 1, for s∈[1,2]. We now argue by contradiction. Suppose the limiting gradien t shrinking soliton ¯gij(s) is ﬂat. Then by (6.2.23), ∇i∇j¯l=1 2s¯gijand ∆ ¯l=n 2s. Putting these into the identity (6.2.21), we get |∇¯l|2=¯l s Since the function ¯lis strictly convex, it follows that√ 4s¯lis a distance function (from some point) on the complete ﬂat manifold ¯M. From the smoothness of the function ¯l, we conclude that the ﬂat manifold ¯Mmust be Rn. In this case we would have its reduced distance to be ¯land its reduced volume to be 1. Since ¯Vis not less than the reduced volume of the limit, this is a contradiction. Theref ore the limiting gradient shrinking soliton ¯ gijis not ﬂat. To conclude this section, we use the above theorem to derive t he classiﬁcation of all two-dimensional ancient κ-solutions which was obtained earlier by Hamilton in Section 26 of [63]. Theorem 6.2.2. The only nonﬂat ancient κ-solutions to Ricci ﬂow on two- dimensional manifolds are the round sphere S2and the round real projective plane RP2. Proof. Letgij(x,t) be a nonﬂat ancient κ-solution deﬁned on M×(−∞,T) (for someT >0), whereMis a two-dimensional manifold. Note that the ancient κ-solution satisﬁes the Li-Yau-Hamilton inequality (Corollary 2.5.5 ). In particular by Corollary 2.5.8, the scalar curvature of the ancient κ-solution is pointwise nondecreasing in 370 H.-D. CAO AND X.-P. ZHU time. Moreover by the strong maximum principle, the ancient κ-solution has strictly positive curvature everywhere. By the above Theorem 6.2.1, we know that the scalings of the an cientκ-solution along a sequence of points qkinMand a sequence of times tk→ −∞ converge to a nonﬂat gradient shrinking soliton ( ¯M,¯gij(x,t)) with −∞<t≤0. We ﬁrst show that the limiting gradient shrinking soliton ( ¯M,¯gij(x,t)) has uni- formly bounded curvature. Clearly, the limiting soliton ha s nonnegative curvature and isκ-noncollapsed on all scales, and its scalar curvature is sti ll pointwise nondecreasing in time. Thus we only need to show that the limiting soliton ha s bounded curvature att= 0. We argue by contradiction. Suppose the curvature of the l imiting soliton is unbounded at t= 0. Of course in this case the limiting soliton ¯Mis noncompact. Then by applying Lemma 6.1.4, we can choose a sequence of poin tsxj,j= 1,2,..., divergent to inﬁnity such that the scalar curvature ¯Rof the limit satisﬁes ¯R(xj,0)≥jand¯R(x,0)≤4¯R(xj,0) for allj= 1,2,..., andx∈B0(xj,j//radicalbig¯R(xj,0)). And then by the nondecreasing (in time) of the scalar curvature, we have ¯R(x,t)≤4¯R(xj,0), for allj= 1,2,...,x∈B0(xj,j//radicalbig¯R(xj,0)) andt≤0. By combining with Hamil- ton’s compactness theorem (Theorem 4.1.5) and the κ-noncollapsing, we know that a subsequence of the rescaling solutions (¯M,¯R(xj,0)¯gij(x,t/¯R(xj,0)),xj), j= 1,2,..., converges in the C∞ loctopology to a nonﬂat smooth solution of the Ricci ﬂow. Then Proposition 6.1.2 implies that the new (two-dimensional) l imit must be ﬂat. This arrives at a contradiction. So we have proved that the limiti ng gradient shrinking soliton has uniformly bounded curvature. We next show that the limiting soliton is compact. Suppose th e limiting soliton is (complete and) noncompact. By the strong maximum princip le we know that the limiting soliton also has strictly positive curvature ever ywhere. After a shift of the time, we may assume that the limiting soliton satisﬁes the fo llowing equation (6.2.24) ∇i∇jf+¯Rij+1 2t¯gij= 0,on− ∞<t< 0, everywhere for some function f. Diﬀerentiating the equation (6.2.24) and switching the order of diﬀerentiations, as in the derivation of (1.1.1 4), we get (6.2.25) ∇i¯R= 2¯Rij∇jf. Fix somet<0, sayt=−1, and consider a long shortest geodesic γ(s), 0≤s≤s. Letx0=γ(0) andX(s) = ˙γ(s). LetV(0) be any unit vector orthogonal to ˙ γ(0) and translateV(0) alongγ(s) to get a parallel vector ﬁeld V(s), 0≤s≤sonγ. Set /hatwideV(s) =  sV(s), for 0≤s≤1, V(s), for 1≤s≤s−1, (s−s)V(s),fors−1≤s≤s. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 371 It follows from the second variation formula of arclength th at /integraldisplays 0(|˙/hatwideV(s)|2−¯R(X,/hatwideV,X,/hatwideV))ds≥0. Thus we clearly have /integraldisplays 0¯R(X,/hatwideV,X,/hatwideV)ds≤const., and then (6.2.26)/integraldisplays 0¯Ric(X,X)ds≤const.. By integrating the equation (6.2.24) we get X(f(γ(s)))−X(f(γ(0))) +/integraldisplays 0¯Ric(X,X)ds−1 2s= 0 and then by (6.2.26), we deduce d ds(f◦γ(s))≥s 2−const., andf◦γ(s)≥s2 4−const.·s−const. fors>0 large enough. Thus at large distances from the ﬁxed point x0the function fhas no critical points and is proper. It then follows from the Morse theory that any two high level sets of fare diﬀeomorphic via the gradient curves of f. Since by (6.2.25), d ds¯R(η(s),−1) =/a\}b∇acketle{t∇¯R,˙η(s)/a\}b∇acket∇i}ht = 2¯Rij∇if∇jf ≥0 for any integral curve η(s) of∇f, we conclude that the scalar curvature ¯R(x,−1) has a positive lower bound on ¯M, which contradicts the Bonnet-Myers Theorem. So we have proved that the limiting gradient shrinking soliton is compact. By Proposition 5.1.10, the compact limiting gradient shrin king soliton has con- stant curvature. This says that the scalings of the ancient κ-solution (M,g ij(x,t)) along a sequence of points qk∈Mand a sequence of times tk→ −∞ converge in the C∞topology to the round S2or the round RP2. In particular, by looking at the time derivative of the volume and the Gauss-Bonnet theorem, we kn ow that the ancient κ-solution (M,g ij(x,t)) exists on a maximal time interval ( −∞,T) withT <+∞. Consider the scaled entropy of Hamilton [60] E(t) =/integraldisplay MRlog[R(T−t)]dVt. 372 H.-D. CAO AND X.-P. ZHU We compute d dtE(t) =/integraldisplay Mlog[R(T−t)]∆RdV t+/integraldisplay M/bracketleftbigg ∆R+R2−R (T−t)/bracketrightbigg dVt (6.2.27) =/integraldisplay M/bracketleftbigg −|∇R|2 R+R2−rR/bracketrightbigg dVt =/integraldisplay M/bracketleftbigg −|∇R|2 R+ (R−r)2/bracketrightbigg dVt wherer=/integraltext MRdV t/Vol t(M) and we have used Vol t(M) = (/integraltext MRdV t)·(T−t) (by the Gauss-Bonnet theorem). For a smooth function fon the surface M, one can readily check /integraldisplay M(∆f)2= 2/integraldisplay M/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇i∇jf−1 2(∆f)gij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/integraldisplay MR|∇f|2, /integraldisplay M|∇R+R∇f|2 R=/integraldisplay M|∇R|2 R−2/integraldisplay MR(∆f) +/integraldisplay MR|∇f|2, and then /integraldisplay M|∇R|2 R+/integraldisplay M(∆f)(∆f−2R) = 2/integraldisplay M/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇i∇jf−1 2(∆f)gij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/integraldisplay M|∇R+R∇f|2 R. By choosing ∆ f=R−r, we get /integraldisplay M|∇R|2 R−/integraldisplay M(R−r)2 = 2/integraldisplay M/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇i∇jf−1 2(∆f)gij/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/integraldisplay M|∇R+R∇f|2 R≥0. If the equality holds, then we have ∇i∇jf−1 2(∆f)gij= 0 i.e.,∇fis conformal. By the Kazdan-Warner identity [77], it follow s that /integraldisplay M∇R· ∇f= 0, so 0 =−/integraldisplay MR∆f =−/integraldisplay M(R−r)2. Hence we have proved the following inequality due to Chow [37 ] (6.2.28)/integraldisplay M|∇R|2 R≥/integraldisplay M(R−r)2, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 373 and the equality holds if and only if R≡r. The combination (6.2.27) and (6.2.28) shows that the scaled entropyE(t) is strictly decreasing along the Ricci ﬂow unless we are on the r ound sphere S2or its quotient RP2. Moreover the convergence result in Theorem 5.1.11 shows th at the scaled entropy Ehas its minimum value at the constant curvature metric (roun dS2 or round RP2). We had shown that the scalings of the nonﬂat ancient κ-solution along a sequence of times tk→ −∞ converge to the constant curvature metric. Then E(t) has its minimal value at t=−∞, so it was constant all along, hence the an- cientκ-solution must have constant curvature for each t∈(−∞,T). This proves the theorem. 6.3. Curvature Estimates via Volume Growth. For solutions to the Ricci ﬂow, Perelman’s no local collapsing theorems tell us that th e local curvature upper bounds imply the local volume lower bounds. Conversely, one would expect to get local curvature upper bounds from local volume lower bounds . If this is the case, one will be able to establish an elliptic type estimate for th e curvatures of solutions to the Ricci ﬂow. This will provide the key estimate for the ca nonical neighborhood structure and thick-thin decomposition of the Ricci ﬂow on t hree-manifolds. In this section we derive such curvature estimates for nonnegative ly curved solutions. In the next chapter we will derive similar estimates for all smo oth solutions, as well as surgically modiﬁed solutions, of the Ricci ﬂow on three-man ifolds. LetMbe ann-dimensional complete noncompact Riemannian manifold wit h nonnegative Ricci curvature. Pick an origin O∈M. The well-known Bishop-Gromov volume comparison theorem tells us the ratio Vol(B(O,r))/rnis monotone nonin- creasing in r∈[0,+∞). Thus there exists a limit νM= lim r→+∞Vol (B(O,r)) rn. Clearly the number νMis invariant under dilation and is independent of the choice of the origin. νMis called the asymptotic volume ratio of the Riemannian manifold M. The following result obtained by Perelman in [103] shows tha t any ancient κ- solution must have zero asymptotic volume ratio. This resul t for the Ricci ﬂow on K¨ ahler manifolds was obtained by Chen and the second author in [32] independently. Moreover in the K¨ ahler case, as shown by Chen, Tang and the se cond author in [29] (for complex two dimension) and by Ni in [98] (for all dimensi ons), the condition of nonnegative curvature operator can be replaced by the weake r condition of nonnega- tive bisectional curvature. Lemma 6.3.1. LetMbe ann-dimensional complete noncompact Riemannian manifold. Suppose gij(x,t),x∈Mandt∈(−∞,T)withT >0, is a nonﬂat ancient solution of the Ricci ﬂow with nonnegative curvature operat or and bounded curvature. Then the asymptotic volume ratio of the solution metric sati sﬁes νM(t) = lim r→+∞Volt(Bt(O,r)) rn= 0 for eacht∈(−∞,T). Proof. The proof is by induction on the dimension. When the dimensio n is two, the lemma is valid by Theorem 6.2.2. For dimension ≥3, we argue by contradiction. 374 H.-D. CAO AND X.-P. ZHU Suppose the lemma is valid for dimensions ≤n−1 and suppose νM(t0)>0 for somen-dimensional nonﬂat ancient solution with nonnegative cur vature operator and bounded curvature at some time t0≤0. Fixing a point x0∈M, we consider the asymptotic scalar curvature ratio A= limsup dt0(x,x0)→+∞R(x,t0)d2 t0(x,x0). We divide the proof into three cases. Case1:A= +∞. By Lemma 6.1.3, there exist sequences of points xk∈Mdivergent to inﬁnity, of radiirk→+∞, and of positive constants δk→0 such that (i)R(x,t0)≤(1+δk)R(xk,t0) for allxin the ballBt0(xk,rk) of radiusrkaround xk, (ii)r2 kR(xk,t0)→+∞ask→+∞, (iii)dt0(xk,x0)/rk→+∞. By scaling the solution around the points xkwith factor R(xk,t0), and shifting the timet0to the new time zero, we get a sequence of rescaled solutions gk(s) =R(xk,t0)g/parenleftbigg ·,t0+s R(xk,t0)/parenrightbigg to the Ricci ﬂow. Since the ancient solution has nonnegative curvature operator and bounded curvature, there holds the Li-Yau-Hamilton ine quality (Corollary 2.5.5). Thus the rescaled solutions satisfy Rk(x,s)≤(1 +δk) for alls≤0 andx∈Bgk(0)(xk,rk/radicalbig R(xk,t0)).SinceνM(t0)>0, it follows from the standard volume comparison and Theorem 4.2.2 that the in jectivity radii of the rescaled solutions gkat the points xkand the new time zero is uniformly bounded be- low by a positive number. Then by Hamilton’s compactness the orem (Theorem 4.1.5), after passing to a subsequence, ( M,g k(s),xk) will converge to a solution ( ˜M,˜g(s),O) to the Ricci ﬂow with ˜R(y,s)≤1,for alls≤0 andy∈˜M, and ˜R(O,0) = 1. Since the metric is shrinking, by (ii) and (iii), we get R(xk,t0)d2 g(·,t0+s R(xk,t0))(x0,xk)≥R(xk,t0)d2 g(·,t0)(x0,xk) which tends to + ∞, ask→+∞, for alls≤0. Thus by Proposition 6.1.2, for eachs≤0, (˜M,˜g(s)) splits oﬀ a line. We now consider the lifting of the solutio n (˜M,˜g(s)),s≤0,to its universal cover and denote it by (˜˜M,˜˜g(s)),s≤0.Clearly we still have ν˜M(0)>0 andν˜˜M(0)>0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 375 By applying Hamilton’s strong maximum principle and the de R ham decomposi- tion theorem, the universal cover˜˜Msplits isometrically as X×Rfor some (n−1)- dimensional nonﬂat (complete) ancient solution Xwith nonnegative curvature op- erator and bounded curvature. These imply that νX(0)>0, which contradicts the induction hypothesis. Case2: 0<A< +∞. Take a sequence of points xkdivergent to inﬁnity such that R(xk,t0)d2 t0(xk,x0)→A,ask→+∞. Consider the rescaled solutions ( M,g k(s)) (around the ﬁxed point x0), where gk(s) =R(xk,t0)g/parenleftbigg ·,t0+s R(xk,t0)/parenrightbigg ,s∈(−∞,0]. Then there is a constant C >0 such that (6.3.1)  Rk(x,0)≤C/d2 k(x,x0,0), Rk(xk,0) = 1, dk(xk,x0,0)→√ A>0, wheredk(·,x0,0) is the distance function from the point x0in the metric gk(0). SinceνM(t0)>0, it is a basic result in Alexandrov space theory (see for exa mple Theorem 7.6 of [20]) that a subsequence of ( M,g k(s),x0) converges in the Gromov- Hausdorﬀ sense to an n-dimensional metric cone ( ˜M,˜g(0),x0) with vertex x0. By (6.3.1), the standard volume comparison and Theorem 4.2. 2, we know that the injectivity radius of ( M,g k(0)) atxkis uniformly bounded from below by a positive numberρ0. After taking a subsequence, we may assume xkconverges to a point x∞ in˜M\{x0}. Then by Hamilton’s compactness theorem (Theorem 4.1.5), w e can take a subsequence such that the metrics gk(s) on the metric balls B0(xk,1 2ρ0)(⊂Mwith respect to the metric gk(0)) converge in the C∞ loctopology to a solution of the Ricci ﬂow on a ball B0(x∞,1 2ρ0). Clearly the C∞ loclimit has nonnegative curvature operator and it is a piece of the metric cone at the time s= 0. By (6.3.1), we have (6.3.2) ˜R(x∞,0) = 1. Letxbe any point in the limiting ball B0(x∞,1 2ρ0) ande1be any radial direction atx. Clearly ˜Ric(e1,e1) = 0. Recall that the evolution equation of the Ricci tensor in frame coordinates is ∂ ∂t˜Rab=˜△˜Rab+ 2˜Racbd˜Rcd. Since the curvature operator is nonnegative, by applying Ha milton’s strong maximum principle (Theorem 2.2.1) to the above equation, we deduce t hat the null space of ˜Ric is invariant under parallel translation. In particular, al l radial directions split oﬀ locally and isometrically. While by (6.3.2), the piece of th e metric cone is nonﬂat. This gives a contradiction. Case3:A= 0. 376 H.-D. CAO AND X.-P. ZHU The gap theorem as was initiated by Mok-Siu-Yau [93] and esta blished by Greene- Wu [49, 50], Eschenberg-Shrader-Strake [45], and Drees [44 ] shows that a complete noncompact n-dimensional (except n= 4 or 8) Riemannian manifold with nonnegative sectional curvature and the asymptotic scalar curvature ra tioA= 0 must be ﬂat. So the present case is ruled out except in dimension n= 4 or 8. Since in our situation the asymptotic volume ratio is positive and the manifold is t he solution of the Ricci ﬂow, we can give an alternative proof for all dimensions as fo llows. We claim the sectional curvature of ( M,g ij(x,t0)) is positive everywhere. Indeed, by Theorem 2.2.2, the image of the curvature operator is just the restricted holonomy algebra Gof the manifold. If the sectional curvature vanishes for som e two-plane, then the holonomy algebra Gcannot beso(n). We observe the manifold is not Einstein since it is noncompact, nonﬂat and has nonnegative curvature oper ator. If Gis irreducible, then by Berger’s Theorem [7], G=u(n 2). Thus the manifold is K¨ ahler with bounded and nonnegative bisectional curvature and with curvature d ecay faster than quadratic. Then by the gap theorem obtained by Chen and the second author in [31], this K¨ ahler manifold must be ﬂat. This contradicts the assumption. Henc e the holonomy algebra Gis reducible and the universal cover of Msplits isometrically as ˜M1×˜M2nontrivially. Clearly the universal cover of Mhas positive asymptotic volume ratio. So ˜M1and ˜M2still have positive asymptotic volume ratio and at least one of them is nonﬂat. By the induction hypothesis, this is also impossible. Thus o ur claim is proved. Now we know that the sectional curvature of ( M,g ij(x,t0)) is positive everywhere. Choose a sequence of points xkdivergent to inﬁnity such that   R(xk,t0)d2 t0(xk,x0) = sup {R(x,t0)d2 t0(x,x0)|dt0(x,x0)≥dt0(xk,x0)}, dt0(xk,x0)≥k, R(xk,t0)d2 t0(xk,x0) =εk→0. Consider the rescaled metric gk(0) =R(xk,t0)g(·,t0) as before. Then (6.3.3)/braceleftigg Rk(x,0)≤εk/d2 k(x,x0,0),fordk(x,x0,0)≥√εk, dk(xk,x0,0) =√εk→0. As in Case 2, the rescaled marked solutions ( M,g k(0),x0) will converge in the Gromov- Hausdorﬀ sense to a metric cone ( ˜M,˜g(0),x0). And by the virtue of Hamilton’s compactness theorem (Theorem 4.1.5), up to a subsequence, t he convergence is in the C∞ loctopology in ˜M\ {x0}. We next claim the metric cone ( ˜M,˜g(0),x0) is isometric toRn. Indeed, let us write the metric cone ˜Mas a warped product R+×rXn−1for some (n−1)-dimensional manifold Xn−1. By (6.3.3), the metric cone must be ﬂat andXn−1is isometric to a quotient of the round sphere Sn−1by ﬁxed point free isometries in the standard metric. To show ˜Mis isometric to Rn, we only need to verify that Xn−1is simply connected. Letϕbe the Busemann function of ( M,g ij(·,t0)) with respect to the point x0. Since (M,g ij(·,t0)) has nonnegative sectional curvature, it is easy to see tha t for any smallε>0, there is a r0>0 such that (1−ε)dt0(x,x0)≤ϕ(x)≤dt0(x,x0) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 377 for allx∈M\Bt0(x0,r0). The strict positivity of the sectional curvature of the manifold (M,g ij(·,t0)) implies that the square of the Busemann function is strict ly convex (and exhausting). Thus every level set ϕ−1(a), witha>inf{ϕ(x)|x∈M}, of the Busemann function ϕis diﬀeomorphic to the ( n−1)-sphere Sn−1. In particular, ϕ−1([a,3 2a]) is simply connected for a>inf{ϕ(x)|x∈M}sincen≥3. Consider an annulus portion [1 ,2]×Xn−1of the metric cone ˜M=R+×rXn−1. It is the limit of ( Mk,gk(0)), where Mk=/braceleftigg x∈M/vextendsingle/vextendsingle/vextendsingle1/radicalbig R(xk,t0)≤dt0(x,x0)≤2/radicalbig R(xk,t0)/bracerightigg . It is clear that ϕ−1/parenleftigg/bracketleftigg 1/radicalbig R(xk,t0),2(1−ε)/radicalbig R(xk,t0)/bracketrightigg/parenrightigg ⊂Mk⊂ϕ−1/parenleftigg/bracketleftigg 1−ε/radicalbig R(xk,t0),2/radicalbig R(xk,t0)/bracketrightigg/parenrightigg forklarge enough. Thus any closed loop in {3 2} ×Xn−1can be shrunk to a point by a homotopy in [1 ,2]×Xn−1. This shows that Xn−1is simply connected. Hence we have proven that the metric cone ( ˜M,˜g(0),x0) is isometric to Rn. Consequently, lim k→+∞Volg(t0)/parenleftbigg Bg(t0)/parenleftbigg x0,r√ R(xk,t0)/parenrightbigg \Bg(t0)/parenleftbigg x0,σr√ R(xk,t0)/parenrightbigg/parenrightbigg /parenleftbigg r√ R(xk,t0)/parenrightbiggn =αn(1−σn) for anyr>0 and 0<σ< 1, whereαnis the volume of the unit ball in the Euclidean space Rn. Finally, by combining with the monotonicity of the Bishop- Gromov volume comparison, we conclude that the manifold ( M,g ij(·,t0)) is ﬂat and isometric to Rn. This contradicts the assumption. Therefore, we have proved the lemma. Finally we would like to include an alternative simpler argu ment, inspired by Ni [98], for the above Case 2 and Case 3 to avoid the use of the gap t heorem, holonomy groups, and asymptotic cone structure. Alternative Proof for Case 2 and Case 3. Let us consider the situation of 0 ≤ A<+∞in the above proof. Observe that νM(t) is nonincreasing in time tby using Lemma 3.4.1(ii) and the fact that the metric is shrinking in t. SupposeνM(t0)>0, then the solution gij(·,t) isκ-noncollapsed for t≤t0for some uniform κ >0. By combining with Theorem 6.2.1, there exist a sequence of poin tsqkand a sequence of timestk→ −∞ such that the scalings of gij(·,t) aroundqkwith factor |tk|−1and with the timestkshifting to the new time zero converge to a nonﬂat gradient sh rinking soliton ¯Min theC∞ loctopology. This gradient soliton also has maximal volume gro wth (i.e.ν¯M(t)>0) and satisﬁes the Li-Yau-Hamilton estimate (Corollary 2. 5.5). If the curvature of the shrinking soliton ¯Mat the time −1 is bounded, then we see from the proof of Theorem 6.2.2 that by using the equations (6.2.2 4)-(6.2.26), the scalar curvature has a positive lower bound everywhere on ¯Mat the time −1. In particular, this implies the asymptotic scalar curvature ratio A=∞for the soliton at the time −1, which reduces to Case 1 and arrives at a contradiction by th e dimension reduction argument. On the other hand, if the scalar curvature is unbou nded, then by Lemma 378 H.-D. CAO AND X.-P. ZHU 6.1.4, the Li-Yau-Hamilton estimate (Corollary 2.5.5) and Lemma 6.1.2, we can do the same dimension reduction as in Case 1 to arrive at a contra diction also. The following lemma is a local and space-time version of Lemm a 6.1.4 on picking local (almost) maximum curvature points. We formulate it fr om Perelman’s argu- ments in the section 10 of [103]. Lemma 6.3.2. For any positive constants B,CwithB >4andC >1000, there exists 1≤A<min{1 4B,1 1000C}which tends to inﬁnity as BandCtend to inﬁnity and satisﬁes the following property. Suppose we have a (not nece ssarily complete) solution gij(t)to the Ricci ﬂow, deﬁned on M×[−t0,0], so that at each time t∈[−t0,0] the metric ball Bt(x0,1)is compactly contained in M. Suppose there exists a point (x′,t′)∈M×(−t0,0]such that dt′(x′,x0)≤1 4and|Rm(x′,t′)|>C+B(t′+t0)−1. Then we can ﬁnd a point (¯x,¯t)∈M×(−t0,0]such that d¯t(¯x,x0)<1 3withQ=|Rm(¯x,¯t)|>C+B(¯t+t0)−1, and |Rm(x,t)| ≤4Q for all (−t0<)¯t−AQ−1≤t≤¯tanddt(x,¯x)≤1 10A1 2Q−1 2. Proof. We ﬁrst claim that there exists a point (¯ x,¯t) with −t0<¯t≤0 and d¯t(¯x,x0)<1 3such that Q=|Rm(¯x,¯t)|>C+B(¯t+t0)−1, and (6.3.4) |Rm(x,t)| ≤4Q wherever ¯t−AQ−1≤t≤¯t, d t(x,x0)≤d¯t(¯x,x0) + (AQ−1)1 2. We will construct such (¯ x,¯t) as a limit of a ﬁnite sequence of points. Take an arbitrary (x1,t1) such that dt1(x1,x0)≤1 4,−t0<t1≤0 and |Rm(x1,t1)|>C+B(t1+t0)−1. Such a point clearly exists by our assumption. Assume we have already constructed (xk,tk).If we cannot take the point ( xk,tk) to be the desired point (¯ x,¯t), then there exists a point ( xk+1,tk+1) such that tk−A|Rm(xk,tk)|−1≤tk+1≤tk, and dtk+1(xk+1,x0)≤dtk(xk,x0) + (A|Rm(xk,tk)|−1)1 2, but |Rm(xk+1,tk+1)|>4|Rm(xk,tk)|. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 379 It then follows that dtk+1(xk+1,x0)≤dt1(x1,x0) +A1 2/parenleftiggk/summationdisplay i=1|Rm(xi,ti)|−1 2/parenrightigg ≤1 4+A1 2/parenleftiggk/summationdisplay i=12−(i−1)|Rm(x1,t1)|−1 2/parenrightigg ≤1 4+ 2(AC−1)1 2 <1 3, tk+1−(−t0) =k/summationdisplay i=1(ti+1−ti) + (t1−(−t0)) ≥ −k/summationdisplay i=1A|Rm(xi,ti)|−1+ (t1−(−t0)) ≥ −Ak/summationdisplay i=14−(i−1)|Rm(x1,t1)|−1+ (t1−(−t0)) ≥ −2A B(t1+t0) + (t1+t0) ≥1 2(t1+t0), and |Rm(xk+1,tk+1)|>4k|Rm(x1,t1)| ≥4kC→+∞ask→+∞. Since the solution is smooth, the sequence {(xk,tk)}is ﬁnite and its last element ﬁts. Thus we have proved assertion (6.3.4). From the above construction we also see that the chosen point (¯x,¯t) satisﬁes d¯t(¯x,x0)<1 3 and Q=|Rm(¯x,¯t)|>C+B(¯t+t0)−1. Clearly, up to some adjustment of the constant A, we only need to show that (6.3.4)′|Rm(x,t)| ≤4Q wherever ¯t−1 200nA1 2Q−1≤t≤¯tanddt(x,¯x)≤1 10A1 2Q−1 2. For any point ( x,¯t) withd¯t(x,¯x)≤1 10A1 2Q−1 2, we have d¯t(x,x0)≤d¯t(¯x,x0) +d¯t(x,¯x) ≤d¯t(¯x,x0) + (AQ−1)1 2 380 H.-D. CAO AND X.-P. ZHU and then by (6.3.4) |Rm(x,¯t)| ≤4Q. Thus by continuity, there is a minimal ¯t′∈[¯t−1 200nA1 2Q−1,¯t] such that (6.3.5) sup/braceleftbigg |Rm(x,t)| |¯t′≤t≤¯t, d t(x,¯x)≤1 10A1 2Q−1 2/bracerightbigg ≤5Q. For any point ( x,t) with ¯t′≤t≤¯tanddt(x,¯x)≤1 10(AQ−1)1 2, we divide the discussion into two cases. Case(1):dt(¯x,x0)≤3 10(AQ−1)1 2. From assertion (6.3.4) we see that (6.3.5)′sup{|Rm(x,t)| |¯t′≤t≤¯t, d t(x,x0)≤(AQ−1)1 2} ≤4Q. Sincedt(¯x,x0)≤3 10(AQ−1)1 2, we have dt(x,x0)≤dt(x,¯x) +dt(¯x,x0) ≤1 10(AQ−1)1 2+3 10(AQ−1)1 2 ≤(AQ−1)1 2 which implies the estimate |Rm(x,t)| ≤4Qfrom (6.3.5)′. Case(2):dt(¯x,x0)>3 10(AQ−1)1 2. From the curvature bounds in (6.3.5) and (6.3.5)′, we can apply Lemma 3.4.1 (ii) withr0=1 10Q−1 2to get d dt(dt(¯x,x0))≥ −40(n−1)Q1 2 and then dt(¯x,x0)≤dˆt(¯x,x0) + 40n(Q1 2)/parenleftbigg1 200nA1 2Q−1/parenrightbigg ≤dˆt(¯x,x0) +1 5(AQ−1)1 2 where ˆt∈(t,¯t] satisﬁes the property that ds(¯x,x0)≥3 10(AQ−1)1 2whenevers∈[t,ˆt]. So we have either dt(x,x0)≤dt(x,¯x) +dt(¯x,x0) ≤1 10(AQ−1)1 2+3 10(AQ−1)1 2+1 5(AQ−1)1 2 ≤(AQ−1)1 2, or dt(x,x0)≤dt(x,¯x) +dt(¯x,x0) ≤1 10(AQ−1)1 2+d¯t(¯x,x0) +1 5(AQ−1)1 2 ≤d¯t(¯x,x0) + (AQ−1)1 2. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 381 It then follows from (6.3.4) that |Rm(x,t)| ≤4Q. Hence we have proved |Rm(x,t)| ≤4Q for any point ( x,t) with ¯t′≤t≤¯tanddt(x,¯x)≤1 10(AQ−1)1 2. By combining with the choice of ¯t′in (6.3.5), we must have ¯t′=¯t−1 200nA1 2Q−1. This proves assertion (6.3.4)′. Therefore we have completed the proof of the lemma. We now use the volume lower bound assumption to establish the crucial curvature upper bound estimate of Perelman [103] for the Ricci ﬂow. For the Ricci ﬂow on K¨ ahler manifolds, a global version of this estimate (i.e., curvature decaying linear in time and quadratic in space) was independently obtained in [ 29] and [32]. Note that the volume estimate conclusion in the following Theorem 6.3 .3 (ii) was not stated in Corollary 11.6 (b) of Perelman [103]. The estimate will be us ed later in the proof of Theorem 7.2.2 and Theorem 7.5.2. Theorem 6.3.3 ( Perelman [103] ).For everyw >0there exist B=B(w)< +∞, C =C(w)<+∞, τ0=τ0(w)>0,andξ=ξ(w)>0 (depending also on the dimension )with the following properties. Suppose we have a (not necessarily complete )solutiongij(t)to the Ricci ﬂow, deﬁned on M×[−t0r2 0,0],so that at each timet∈[−t0r2 0,0]the metric ball Bt(x0,r0)is compactly contained in M. (i) If at each time t∈[−t0r2 0,0], Rm(.,t)≥ −r−2 0onBt(x0,r0) andVolt(Bt(x0,r0))≥wrn 0, then we have the estimate |Rm(x,t)| ≤Cr−2 0+B(t+t0r2 0)−1 whenever −t0r2 0<t≤0anddt(x,x0)≤1 4r0. (ii) If for some 0<¯τ≤t0, Rm(x,t)≥ −r−2 0fort∈[−¯τr2 0,0],x∈Bt(x0,r0), andVol0(B0(x0,r0))≥wrn 0, then we have the estimates Volt(Bt(x0,r0))≥ξrn 0for all max{−¯τr2 0,−τ0r2 0} ≤t≤0, and |Rm(x,t)| ≤Cr−2 0+B(t−max{−¯τr2 0,−τ0r2 0})−1 whenever max{−¯τr2 0,−τ0r2 0}<t≤0anddt(x,x0)≤1 4r0. 382 H.-D. CAO AND X.-P. ZHU Proof. By scaling we may assume r0= 1. (i) By the standard (relative) volume comparison, we know th at there exists somew′>0, withw′≤w, depending only on w, such that for each point ( x,t) with −t0≤t≤0 anddt(x,x0)≤1 3,and for each r≤1 3, there holds (6.3.6) Vol t(Bt(x,r))≥w′rn. We argue by contradiction. Suppose there are sequences B,C→+∞,of solutions gij(t) and points ( x′,t′) such that dt′(x′,x0)≤1 4,−t0<t′≤0 and |Rm(x′,t′)|>C+B(t′+t0)−1. Then by Lemma 6.3.2, we can ﬁnd a sequence of points (¯ x,¯t) such that d¯t(¯x,x0)<1 3, Q=|Rm(¯x,¯t)|>C+B(¯t+t0)−1, and |Rm(x,t)| ≤4Q wherever ( −t0<)¯t−AQ−1≤t≤¯t, d t(x,¯x)≤1 10A1 2Q−1 2, whereAtends to inﬁnity withB,C. Thus we may take a blow-up limit along the points (¯ x,¯t) with factors Q and get a non-ﬂat ancient solution ( M∞,g(∞) ij(t)) with nonnegative curvature operator and with the asymptotic volume ratio νM∞(t)≥w′>0 for eacht∈(−∞,0] (by (6.3.6)). This contradicts Lemma 6.3.1. (ii) LetB(w), C(w) be good for the ﬁrst part of the theorem. By the volume assumption at t= 0 and the standard (relative) volume comparison, we still h ave the estimate (6.3.6)′Vol0(B0(x,r))≥w′rn for eachx∈Mwithd0(x,x0)≤1 3andr≤1 3. We will show that ξ= 5−nw′, B=B(5−nw′) andC=C(5−nw′) are good for the second part of the theorem. By continuity and the volume assumption at t= 0, there is a maximal subinterval [−τ,0] of the time interval [ −¯τ,0] such that Volt(Bt(x0,1))≥w≥5−nw′for allt∈[−τ,0]. This says that the assumptions of (i) hold with 5−nw′in place of wand withτin place oft0. Thus the conclusion of the part (i) gives us the estimate (6.3.7) |Rm(x,t)| ≤C+B(t+τ)−1 whenevert∈(−τ,0] anddt(x,x0)≤1 4. We need to show that one can choose a positive τ0depending only on wand the dimension such that the maximal τ≥min{¯τ,τ0}. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 383 Fort∈(−τ,0] and1 8≤dt(x,x0)≤1 4, we use (6.3.7) and Lemma 3.4.1(ii) to get d dtdt(x,x0)≥ −10(n−1)(√ C+ (√ B/√ t+τ)) which further gives d0(x,x0)≥d−τ(x,x0)−10(n−1)(τ√ C+ 2√ Bτ). This means (6.3.8) B(−τ)(x0,1 4)⊃B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg . Note that the scalar curvature R≥ −C(n) for some constant C(n) depending only on the dimension since Rm≥ −1.We have d dtVolt/parenleftbigg B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg/parenrightbigg =/integraldisplay B0(x0,1 4−10(n−1)(τ√ C+2√ Bτ))(−R)dVt ≤C(n)Vol t/parenleftbigg B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg/parenrightbigg and then Volt/parenleftbigg B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg/parenrightbigg (6.3.9) ≤eC(n)τVol(−τ)/parenleftbigg B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg/parenrightbigg . Thus by (6.3.6)′, (6.3.8) and (6.3.9), Vol(−τ)(B(−τ))(x0,1) ≥Vol(−τ)(B(−τ))/parenleftbigg x0,1 4/parenrightbigg ≥Vol(−τ)/parenleftbigg B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg/parenrightbigg ≥e−C(n)τVol0/parenleftbigg B0/parenleftbigg x0,1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbigg/parenrightbigg ≥e−C(n)τw′/parenleftbigg1 4−10(n−1)(τ√ C+ 2√ Bτ)/parenrightbiggn . So it suﬃces to choose τ0=τ0(w) small enough so that e−C(n)τ0/parenleftbigg1 4−10(n−1)(τ0√ C+ 2/radicalbig Bτ0)/parenrightbiggn ≥/parenleftbigg1 5/parenrightbiggn . Therefore we have proved the theorem. 384 H.-D. CAO AND X.-P. ZHU 6.4. Ancient κ-solutions on Three-manifolds. In this section we will deter- mine the structures of ancient κ-solutions on three-manifolds. First of all, we consider a special class of ancient solution s — gradient shrinking Ricci solitons. Recall that a solution gij(t) to the Ricci ﬂow is said to be a gradient shrinking Ricci soliton if there exists a smooth function fsuch that (6.4.1) ∇i∇jf+Rij+1 2tgij= 0 for − ∞<t< 0. A gradient shrinking Ricci soliton moves by the one paramete r group of diﬀeomor- phisms generated by ∇fand shrinks by a factor at the same time. The following result of Perelman [103] gives a complete clas siﬁcation for all three- dimensional complete κ-noncollapsed gradient shrinking solitons with bounded an d nonnegative sectional curvature. Lemma 6.4.1 ( Classiﬁcation of three-dimensional shrinking solitons ).Let (M,g ij(t))be a nonﬂat gradient shrinking soliton on a three-manifold. Suppose (M,g ij(t))has bounded and nonnegative sectional curvature and is κ-noncollapsed on all scales for some κ>0. Then (M,g ij(t))is one of the following: (i) the round three-sphere S3, or a metric quotient of S3; (ii) the round inﬁnite cylinder S2×R, or one of its Z2quotients. Proof. We ﬁrst consider the case that the sectional curvature of the nonﬂat gradient shrinking soliton is not strictly positive. Let us pull back the soliton to its universal cover. Then the pull-back metric is again a nonﬂat ancientκ-solution. By Hamilton’s strong maximum principle (Theorem 2.2.1), we kn ow that the pull-back solution splits as the metric product of a two-dimensional n onﬂat ancient κ-solution andR. Since the two-dimensional nonﬂat ancient κ-solution is simply connected, it follows from Theorem 6.2.2 that it must be the round sphere S2. Thus, the gradient shrinking soliton must be S2×R/Γ, a metric quotient of the round cylinder. For eachσ∈Γ and (x,s)∈S2×R, we write σ(x,s) = (σ1(x,s),σ2(x,s))∈ S2×R. Sinceσsends lines to lines, and sends cross spheres to cross sphere s, we have σ2(x,s) =σ2(y,s), for allx,y∈S2. This says that σ2reduces to a function of s alone on R. Moreover, for any ( x,s),(x′,s′)∈S2×R, sinceσpreserves the distances between cross spheres S2×{s}andS2×{s′}, we have |σ2(x,s)−σ2(x′,s′)|=|s−s′|. So the projection Γ 2of Γ to the second factor Ris an isometry subgroup of R. If the metric quotient S2×R/Γ were compact, it would not be κ-noncollapsed on suﬃciently large scales as t→ −∞ . Thus the metric quotient S2×R/Γ is noncompact. It follows that Γ 2={1}orZ2. In particular, there is a Γ-invariant cross sphere S2in the round cylinder S2×R. Denote it by S2×{0}. Then Γ acts on the round two-sphere S2×{0} isometrically without ﬁxed points. This implies Γ is either {1}orZ2. Hence we conclude that the gradient shrinking soliton is either the r ound cylinder S2×R, or RP2×R, or the twisted product S2˜×Rwhere Z2ﬂips both S2andR. We next consider the case that the gradient shrinking solito n is compact and has strictly positive sectional curvature everywhere. By the p roof of Theorem 5.2.1 (see also Remark 5.2.8) we see that the compact gradient shrinkin g soliton is getting round and tends to a space form (with positive constant curvature) as the time approaches the maximal time t= 0. Since the shape of a gradient shrinking Ricci soliton doe s not change up to reparametrizations and homothetical scali ngs, the gradient shrinking soliton has to be the round three-sphere S3or a metric quotient of S3. Finally we want to exclude the case that the gradient shrinki ng soliton is non- compact and has strictly positive sectional curvature ever ywhere. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 385 Suppose there is a (complete three-dimensional) noncompac tκ-noncollapsed gra- dient shrinking soliton gij(t),−∞<t< 0, with bounded and positive sectional cur- vature at each t∈(−∞,0) and satisfying the equation (6.4.1). Then as in (6.2.25), we have (6.4.2) ∇iR= 2Rij∇jf. Fix somet<0, sayt=−1, and consider a long shortest geodesic γ(s), 0≤s≤¯s. Letx0=γ(0) andX(s) = ˙γ(s).LetU(0) be any unit vector orthogonal to ˙ γ(0) and translateU(0) alongγ(s) to get a parallel vector ﬁeld U(s), 0≤s≤¯s, onγ. Set /tildewideU(s) =  sU(s), for 0 ≤s≤1, U(s), for 1 ≤s≤¯s−1 (¯s−s)U(s),for ¯s−1≤s≤¯s. It follows from the second variation formula of arclength th at /integraldisplay¯s 0(|˙/tildewideU(s)|2−R(X,/tildewideU,X,/tildewideU))ds≥0. Since the curvature of the metric gij(−1) is bounded, we clearly have /integraldisplay¯s 0R(X,U,X,U )ds≤const. and then (6.4.3)/integraldisplay¯s 0Ric (X,X)ds≤const.. Moreover, since the curvature of the metric gij(−1) is positive, it follows from the Cauchy-Schwarz inequality that for any unit vector ﬁeld Yalongγand orthogonal to X(= ˙γ(s)),we have /integraldisplay¯s 0|Ric (X,Y)|2ds≤/integraldisplay¯s 0Ric (X,X)Ric(Y,Y)ds ≤const.·/integraldisplay¯s 0Ric (X,X)ds ≤const. and then (6.4.4)/integraldisplay¯s 0|Ric (X,Y)|ds≤const.·(√ ¯s+ 1). From (6.4.1) we know ∇X∇Xf+ Ric(X,X)−1 2= 0 and by integrating this equation we get X(f(γ(¯s)))−X(f(γ(0))) +/integraldisplay¯s 0Ric (X,X)ds−1 2¯s= 0. 386 H.-D. CAO AND X.-P. ZHU Thus by (6.4.3) we deduce (6.4.5)¯s 2−const.≤ /a\}b∇acketle{tX,∇f(γ(¯s))/a\}b∇acket∇i}ht ≤¯s 2+ const.. Similarly by integrating (6.4.1) and using (6.4.4) we can de duce (6.4.6) |/a\}b∇acketle{tY,∇f(γ(¯s))/a\}b∇acket∇i}ht| ≤const.·(√ ¯s+ 1). These two inequalities tell us that at large distances from t he ﬁxed point x0the functionfhas no critical point, and its gradient makes a small angle wi th the gradient of the distance function from x0. Now from (6.4.2) we see that at large distances from x0,Ris strictly increasing along the gradient curves of f, in particular ¯R= limsup d(−1)(x,x0)→+∞R(x,−1)>0. Let us choose a sequence of points ( xk,−1) whereR(xk,−1)→¯R. By the noncol- lapsing assumption we can take a limit along this sequence of points of the gradient soliton and get an ancient κ-solution deﬁned on −∞<t< 0. By Proposition 6.1.2, we deduce that the limiting ancient κ-solution splits oﬀ a line. Since the soliton has positive sectional curvature, we know from Gromoll-Meyer [ 52] that it is orientable. Then it follows from Theorem 6.2.2 that the limiting solutio n is the shrinking round inﬁnite cylinder with scalar curvature ¯Rat timet=−1. Since the limiting solution exists on ( −∞,0), we conclude that ¯R≤1. Hence R(x,−1)<1 when the distance from xto the ﬁxed x0is large enough on the gradient shrinking soliton. Let us consider the level surface {f=a}off. The second fundamental form of the level surface is given by hij=/angbracketleftbigg ∇i/parenleftbigg∇f |∇f|/parenrightbigg ,ej/angbracketrightbigg =∇i∇jf/|∇f|, i,j = 1,2, where {e1,e2}is an orthonormal basis of the level surface. By (6.4.1), we h ave ∇ei∇eif=1 2−Ric (ei,ei)≥1 2−R 2>0, i = 1,2, since for a three-manifold the positivity of sectional curv ature is equivalent to R≥ 2Ric. It then follows from the ﬁrst variation formula that d daArea{f=a}=/integraldisplay {f=a}div/parenleftbigg∇f |∇f|/parenrightbigg (6.4.7) ≥/integraldisplay {f=a}1 |∇f|(1−R) >/integraldisplay {f=a}1 |∇f|(1−¯R) ≥0 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 387 foralarge enough. We conclude that Area {f=a}strictly increases as aincreases. From (6.4.5) we see that for slarge enough /vextendsingle/vextendsingle/vextendsingle/vextendsingledf ds−s 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤const., and then /vextendsingle/vextendsingle/vextendsingle/vextendsinglef−s2 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤const.·(s+ 1). Thus we get from (6.4.7) d daArea{f=a}>1−¯R 2√aArea{f=a} foralarge enough. This implies that log Area {f=a}>(1−¯R)√a−const. foralarge enough. But it is clear from (6.4.7) that Area {f=a}is uniformly bounded from above by the area of the round sphere of scalar curvature ¯Rfor all large a. Thus we deduce that ¯R= 1. So (6.4.8) Area {f=a}<8π foralarge enough. Denote byXthe unit normal vector to the level surface {f=a}. By using the Gauss equation and (6.4.1), the intrinsic curvature of the l evel surface {f=a}can be computed as intrinsic curvature (6.4.9) =R1212+ det(hij) =1 2(R−2Ric (X,X)) +det(Hess (f)) |∇f|2 ≤1 2(R−2Ric (X,X)) +1 4|∇f|2(tr (Hess (f)))2 =1 2(R−2Ric (X,X)) +1 4|∇f|2(1−(R−Ric (X,X)))2 =1 2/bracketleftbigg 1−Ric (X,X)−(1−R+ Ric(X,X)) +(1−R+ Ric (X,X))2 2|∇f|2/bracketrightbigg <1 2 for suﬃciently large a, since (1 −R+Ric(X,X))>0 and|∇f|is large when ais large. Thus the combination of (6.4.8) and (6.4.9) gives a contradi ction to the Gauss-Bonnet formula. Therefore we have proved the lemma. As a direct consequence, there is a universal positive const antκ0such that any nonﬂat three-dimensional gradient shrinking soliton, which is also an ancient 388 H.-D. CAO AND X.-P. ZHU κ-solution, to the Ricci ﬂow must be κ0-noncollapsed on all scales unless it is a met- ric quotient of round three-sphere. The following result, c laimed by Perelman in the section 1.5 of [104], shows that this property actually h olds for all nonﬂat three- dimensional ancient κ-solutions. Proposition 6.4.2 ( Universal noncollapsing ).There exists a positive constant κ0with the following property. Suppose we have a nonﬂat three- dimensional ancient κ-solution for some κ>0. Then either the solution is κ0-noncollapsed on all scales, or it is a metric quotient of the round three-sphere. Proof. Letgij(x,t),x∈Mandt∈(−∞,0], be a nonﬂat ancient κ-solution for someκ>0. For an arbitrary point ( p,t0)∈M×(−∞,0], we deﬁne as in Chapter 3 that τ=t0−t,fort<t 0, l(q,τ) =1 2√τinf/braceleftbigg/integraldisplayτ 0√s(R(γ(s),t0−s) +|˙γ(s)|2 gij(t0−s))ds| γ: [0,τ]→Mwithγ(0) =p,γ(τ) =q/bracerightbigg , and/tildewideV(τ) =/integraldisplay M(4πτ)−3 2exp(−l(q,τ))dVt0−τ(q). Recall from (6.2.1) that for each τ >0 we can ﬁnd q=q(τ) such that l(q,τ)≤3 2. In view of Lemma 6.4.1, we may assume that the ancient κ-solution is not a gradient shrinking Ricci soliton. Thus by (the proof of) Theorem 6.2. 1, the scalings of gij(t0−τ) atq(τ) with factor τ−1converge along a subsequence of τ→+∞to a nonﬂat gradient shrinking soliton with nonnegative curvature operator whi ch isκ-noncollapsed on all scales. We now show that the limit has bounded curvature. Denote the limiting nonﬂat gradient shrinking soliton by ( ¯M,¯gij(x,t)) with −∞< t≤0. Note that there holds the Li-Yau-Hamilton inequality (Th eorem 2.5.4) on any ancientκ-solution and in particular, the scalar curvature of the anc ientκ-solution is pointwise nondecreasing in time. This implies that the scal ar curvature of the limiting soliton ( ¯M,¯gij(x,t)) is still pointwise nondecreasing in time. Thus we only nee d to show that the limiting soliton has bounded curvature at t= 0. We argue by contradiction. By lifting to its orientable cove r, we may assume that ¯Mis orientable. Suppose the curvature of the limiting solito n is unbounded at t= 0. Of course in this case the limiting soliton ¯Mis noncompact. Then by applying Lemma 6.1.4, we can choose a sequence of points xj,j= 1,2,...,divergent to inﬁnity such that the scalar curvature ¯Rof the limit satisﬁes ¯R(xj,0)≥jand¯R(x,0)≤4¯R(xj,0) for allx∈B0(xj,j//radicalbig¯R(xj,0)) andj= 1,2,.... Since the scalar curvature is nonde- creasing in time, we have (6.4.10) ¯R(x,t)≤4¯R(xj,0), for allx∈B0(xj,j//radicalbig¯R(xj,0)), allt≤0 andj= 1,2,.... By combining with Hamilton’s compactness theorem (Theorem 4.1.5) and the κ-noncollapsing, we know that a subsequence of the rescaled solutions (¯M,¯R(xj,0)¯gij(x,t/¯R(xj,0)),xj), j= 1,2,..., THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 389 converges in the C∞ loctopology to a nonﬂat smooth solution of the Ricci ﬂow. Then Proposition 6.1.2 implies that the new limit at the new time {t= 0}must split oﬀ a line. By pulling back the new limit to its universal cover an d applying Hamilton’s strong maximum principle, we deduce that the pull-back of th e new limit on the universal cover splits oﬀ a line for all time t≤0. Thus by combining with Theorem 6.2.2 and the argument in the proof of Lemma 6.4.1, we further deduce that the new limit is either the round cylinder S2×Ror the round RP2×R. Since ¯Mis orientable, the new limit must be S2×R. Since ( ¯M,¯gij(x,0)) has nonnegative curvature operator and the points {xj}going to inﬁnity and ¯R(xj,0)→+∞, this gives a contradiction to Proposition 6.1.1. So we have proved that the limiting gra dient shrinking soliton has bounded curvature at each time. Hence by Lemma 6.4.1, the limiting gradient shrinking solit on is either the round three-sphere S3or its metric quotients, or the inﬁnite cylinder S2×Ror one of its Z2quotients. If the asymptotic gradient shrinking soliton is the round three-sphere S3or its metric quotients, it follows from Lemma 5.2.4 and Prop osition 5.2.5 that the ancient κ-solution must be round. Thus in the following we may assume t he asymptotic gradient shrinking soliton is the inﬁnite cylin derS2×Ror aZ2quotient ofS2×R. We now come back to consider the original ancient κ-solution (M,g ij(x, t)). By rescaling, we can assume that R(x,t)≤1 for all (x,t) satisfying dt0(x,p)≤2 and t∈[t0−1,t0]. We will argue as in the proof of Theorem 3.3.2 (Perelman’s n o local collapsing theorem I) to obtain a positive lower bound for Vo lt0(Bt0(p,1)). Denote by ξ= Vol t0(Bt0(p,1))1 3. For anyv∈TpMwe can ﬁnd an L-geodesic γ(τ), starting at p, with lim τ→0+√τ˙γ(τ) =v.It follows from the L-geodesic equation (3.2.1) that d dτ(√τ˙γ)−1 2√τ∇R+ 2Ric (√τ˙γ,·) = 0. By integrating as before we see that for τ≤ξwith the property γ(σ)∈Bt0(p,1) as long asσ<τ, there holds |√τ˙γ(τ)−v| ≤Cξ(|v|+ 1) whereCis some positive constant depending only on the dimension. W ithout loss of generality, we may assume Cξ≤1 4andξ≤1 100. Then for v∈TpMwith|v| ≤1 4ξ−1 2 and forτ≤ξwith the property γ(σ)∈Bt0(p,1) as long as σ<τ, we have dt0(p,γ(τ))≤/integraldisplayτ 0|˙γ(σ)|dσ <1 2ξ−1 2/integraldisplayτ 0dσ√σ = 1. This shows (6.4.11) Lexp/braceleftbigg |v| ≤1 4ξ−1 2/bracerightbigg (ξ)⊂Bt0(p,1). 390 H.-D. CAO AND X.-P. ZHU We decompose Perelman’s reduced volume /tildewideV(ξ) as /tildewideV(ξ) =/integraldisplay Lexp /D2 |v|≤1 4ξ−1 2 /D3 (ξ)(6.4.12) +/integraldisplay M\Lexp /D2 |v|≤1 4ξ−1 2 /D3 (ξ)(4πξ)−3 2exp(−l(q,ξ))dVt0−ξ(q). By using (6.4.11) and the metric evolution equation of the Ri cci ﬂow, the ﬁrst term on the RHS of (6.4.12) can be estimated by /integraldisplay Lexp{|v|≤1 4ξ−1 2}(ξ)(4πξ)−3 2exp(−l(q,ξ))dVt0−ξ(q) ≤/integraldisplay Bt0(p,1)(4πξ)−3 2e3ξdVt0(q) = (4π)−3 2e3ξξ3 2 <ξ3 2, while by using Theorem 3.2.7 (Perelman’s Jacobian comparis on theorem), the second term on the RHS of (6.4.12) can be estimated by /integraldisplay M\Lexp /D2 |v|≤1 4ξ−1 2 /D3 (ξ)(4πξ)−3 2exp(−l(q,ξ))dVt0−ξ(q) (6.4.13) ≤/integraldisplay {|v|>1 4ξ−1 2}(4πτ)−3 2exp(−l(τ))J(τ)|τ=0dv = (4π)−3 2/integraldisplay {|v|>1 4ξ−1 2}exp(−|v|2)dv <ξ3 2 since lim τ→0+τ−3 2J(τ) = 1 and lim τ→0+l(τ) =|v|2by (3.2.18) and (3.2.19) respec- tively. Thus we obtain (6.4.14) /tildewideV(ξ)<2ξ3 2. On the other hand, we recall that there exist a sequence τk→+∞and a sequence of pointsq(τk)∈Mwithl(q(τk),τk)≤3 2so that the scalings of the ancient κ-solution atq(τk) with factor τ−1 kconverge to either round S2×Ror one of its Z2quotients. For suﬃciently large k, we construct a path γ: [0,2τk]→M, connecting pto any given point q∈M, as follows: the ﬁrst half path γ|[0,τk]connectsptoq(τk) such that l(q(τk),τk) =1 2√τk/integraldisplayτk 0√τ(R+|˙γ(τ)|2)dτ≤2, and the second half path γ|[τk,2τk]is a shortest geodesic connecting q(τk) toqwith respect to the metric gij(t0−τk). Note that the rescaled metric τ−1 kgij(t0−τ) over the domain Bt0−τk(q(τk),√τk)×[t0−2τk,t0−τk] is suﬃciently close to the round THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 391 S2×Ror its Z2quotients. Then there is a universal positive constant βsuch that l(q,2τk)≤1 2√2τk/parenleftbigg/integraldisplayτk 0+/integraldisplay2τk τk/parenrightbigg√τ(R+|˙γ(τ)|2)dτ ≤√ 2 +1 2√2τk/integraldisplay2τk τk√τ(R+|˙γ(τ)|2)dτ ≤β for allq∈Bt0−τk(q(τk),√τk). Thus /tildewideV(2τk) =/integraldisplay M(4π(2τk))−3 2exp(−l(q,2τk))dVt0−2τk(q) ≥e−β/integraldisplay Bt0−τk(q(τk),√τk)(4π(2τk))−3 2dVt0−2τk(q) ≥˜β for some universal positive constant ˜β. Here we have used the curvature estimate (6.2.6). By combining with the monotonicity of Perelman’s r educed volume (Theorem 3.2.8) and (6.4.14), we deduce that ˜β≤/tildewideV(2τk)≤/tildewideV(ξ)<2ξ3 2. This proves Volt0(Bt0(p,1))≥κ0>0 for some universal positive constant κ0. So we have proved that the ancient κ-solution is also an ancient κ0-solution. The important Li-Yau-Hamilton inequality gives rise to a pa rabolic Harnack es- timate (Corollary 2.5.7) for solutions of the Ricci ﬂow with bounded and nonnegative curvature operator. As explained in the previous section, t he no local collapsing the- orem of Perelman implies a volume lower bound from a curvatur e upper bound, while the estimate in the previous section implies a curvature upp er bound from a volume lower bound. The combination of these two estimates as well a s the Li-Yau-Hamilton inequality will give an important elliptic type property fo r three-dimensional ancient κ-solutions. This elliptic type property was ﬁrst implicitl y given by Perelman in [103] and it will play a crucial role in the analysis of singulariti es. Theorem 6.4.3 ( Elliptic type estimate ).There exist a positive constant ηand a positive increasing function ω: [0,+∞)→(0,+∞)with the following properties. Suppose we have a three-dimensional ancient κ-solution (M,g ij(t)),−∞<t≤0,for someκ>0. Then (i) for every x,y∈Mandt∈(−∞,0], there holds R(x,t)≤R(y,t)·ω(R(y,t)d2 t(x,y)); (ii) for all x∈Mandt∈(−∞,0], there hold |∇R|(x,t)≤ηR3 2(x,t)and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂R ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x,t)≤ηR2(x,t). 392 H.-D. CAO AND X.-P. ZHU Proof. (i) Consider a three-dimensional nonﬂat ancient κ-solutiongij(x,t) on M×(−∞,0]. In view of Proposition 6.4.2, we may assume that the ancie nt solution is universalκ0-noncollapsed. Obviously we only need to establish the esti mate att= 0. Letybe an arbitrarily ﬁxed point in M. By rescaling, we can assume R(y,0) = 1. Let us ﬁrst consider the case that sup {R(x,0)d2 0(x,y)|x∈M}>1. Deﬁnezto be the closest point to y(at timet= 0) satisfying R(z,0)d2 0(z,y) = 1. We want to boundR(x,0)/R(z,0) from above for x∈B0(z,2R(z,0)−1 2). Connectyandzby a shortest geodesic and choose a point ˜ zlying on the geo- desic satisfying d0(˜z,z) =1 4R(z,0)−1 2. Denote by Bthe ball centered at ˜ zand with radius1 4R(z,0)−1 2(with respect to the metric at t= 0). Clearly the ball Blies in B0(y,R(z,0)−1 2) and lies outside B0(y,1 2R(z,0)−1 2). Thus for x∈B, we have R(x,0)d2 0(x,y)≤1 andd0(x,y)≥1 2R(z,0)−1 2 and hence R(x,0)≤1 (1 2R(z,0)−1 2)2for allx∈B. Then by the Li-Yau-Hamilton inequality and the κ0-noncollapsing, we have Vol0(B)≥κ0/parenleftbigg1 4R(z,0)−1 2/parenrightbigg3 , and then Vol0(B0(z,8R(z,0)−1 2))≥κ0 215(8R(z,0)−1 2)3. So by Theorem 6.3.3(ii), there exist positive constants B(κ0),C(κ0),andτ0(κ0) such that (6.4.15) R(x,0)≤(C(κ0) +B(κ0) τ0(κ0))R(z,0) for allx∈B0(z,2R(z,0)−1 2). We now consider the remaining case. If R(x,0)d2 0(x,y)≤1 everywhere, we choose a pointzsatisfying sup {R(x,0)|x∈M} ≤2R(z,0). Obviously we also have the estimate (6.4.15) in this case. We next want to bound R(z,0) for the chosen z∈M. By (6.4.15) and the Li-Yau-Hamilton inequality, we have R(x,t)≤(C(κ0) +B(κ0) τ0(κ0))R(z,0) for allx∈B0(z,2R(z,0)−1 2) and allt≤0. It then follows from the local derivative estimates of Shi that ∂R ∂t(z,t)≤/tildewideC(κ0)R(z,0)2,for all −R−1(z,0)≤t≤0 which implies (6.4.16) R(z,−cR−1(z,0))≥cR(z,0) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 393 for some small positive constant cdepending only on κ0. On the other hand, by using the Harnack estimate in Corollary 2.5.7, we have (6.4.17) 1 = R(y,0)≥/tildewidecR(z,−cR−1(z,0)) for some small positive constant /tildewidecdepending only on κ0, sinced0(y,z)≤R(z,0)−1 2 and the metric gij(t) is equivalent on B0(z,2R(z,0)−1 2)×[−cR−1(z,0),0] withc>0 small enough. Thus we get from (6.4.16) and (6.4.17) that (6.4.18) R(z,0)≤/tildewideA for some positive constant /tildewideAdepending only on κ0. SinceB0(z,2R(z,0)−1 2)⊃B0(y,R(z,0)−1 2) andR(z,0)−1 2≥(/tildewideA)−1 2, the combi- nation of (6.4.15) and (6.4.18) gives (6.4.19) R(x,0)≤(C(κ0) +B(κ0) τ0(κ0))/tildewideA wheneverx∈B0(y,(/tildewideA)−1 2). Then by the κ0-noncollapsing there exists a positive constantr0depending only on κ0such that Vol0(B0(y,r0))≥κ0r3 0. For any ﬁxed R0≥r0, we then have Vol0(B0(y,R0))≥κ0r3 0=κ0(r0 R0)3·R3 0. By applying Theorem 6.3.3 (ii) again and noting that the cons tantκ0is universal, there exists a positive constant ω(R0) depending only on R0such that R(x,0)≤ω(R2 0) for all x∈B0(y,1 4R0). This gives the desired estimate. (ii) This follows immediately from conclusion (i), the Li-Y au-Hamilton inequality and the local derivative estimate of Shi. As a consequence, we have the following compactness result d ue to Perelman [103]. Corollary 6.4.4 ( Compactness of ancient κ0-solutions ).The set of nonﬂat three-dimensional ancient κ0-solutions is compact modulo scaling in the sense that for any sequence of such solutions and marking points (xk,0)withR(xk,0) = 1 , we can extract a C∞ locconverging subsequence whose limit is also an ancient κ0-solution. Proof. Consider any sequence of three-dimensional ancient κ0-solutions and mark- ing points ( xk,0) withR(xk,0) = 1. By Theorem 6.4.3(i), the Li-Yau-Hamilton inequality and Hamilton’s compactness theorem (Theorem 4. 1.5), we can extract a C∞ locconverging subsequence such that the limit ( ¯M,¯gij(x,t)), with −∞< t≤0, is an ancient solution to the Ricci ﬂow with nonnegative curv ature operator and κ0- noncollapsed on all scales. Since any ancient κ0-solution satisﬁes the Li-Yau-Hamilton 394 H.-D. CAO AND X.-P. ZHU inequality, it implies that the scalar curvature ¯R(x,t) of the limit ( ¯M,¯gij(x,t)) is pointwise nondecreasing in time. Thus it remains to show tha t the limit solution has bounded curvature at t= 0. Obviously we may assume the limiting manifold ¯Mis noncompact. By pulling back the limiting solution to its orientable cover, we can as sume that the limiting man- ifold ¯Mis orientable. We now argue by contradiction. Suppose the sc alar curvature ¯Rof the limit at t= 0 is unbounded. By applying Lemma 6.1.4, we can choose a sequence of points xj∈¯M,j = 1,2,...,divergent to inﬁnity such that the scalar curvature ¯Rof the limit satisﬁes ¯R(xj,0)≥jand¯R(x,0)≤4¯R(xj,0) for allj= 1,2,...,andx∈B0(xj,j//radicalbig¯R(xj,0)). Then from the fact that the limiting scalar curvature ¯R(x,t) is pointwise nondecreasing in time, we have (6.4.20) ¯R(x,t)≤4¯R(xj,0) for allj= 1,2,...,x∈B0(xj,j//radicalbig¯R(xj,0)) andt≤0. By combining with Hamilton’s compactness theorem (Theorem 4.1.5) and the κ0-noncollapsing, we know that a subsequence of the rescaled solutions (¯M,¯R(xj,0)¯gij(x,t/¯R(xj,0)),xj), j= 1,2,..., converges in the C∞ loctopology to a nonﬂat smooth solution of the Ricci ﬂow. Then Proposition 6.1.2 implies that the new limit at the new time {t= 0}must split oﬀ a line. By pulling back the new limit to its universal cover an d applying Hamilton’s strong maximum principle, we deduce that the pull-back of th e new limit on the universal cover splits oﬀ a line for all time t≤0. Thus by combining with Theorem 6.2.2 and the argument in the proof of Lemma 6.4.1, we further deduce that the new limit is either the round cylinder S2×Ror the round RP2×R. Since ¯Mis orientable, the new limit must be S2×R. Moreover, since ( ¯M,¯gij(x,0)) has nonnegative curvature operator and the points {xj}are going to inﬁnity and ¯R(xj,0)→+∞, this gives a contradiction to Proposition 6.1.1. So we have proved that t he limit ( ¯M,¯gij(x,t)) has uniformly bounded curvature. Arbitrarily ﬁx ε >0. Letgij(x,t) be a nonﬂat ancient κ-solution on a three- manifoldMfor someκ >0. We say that a point x0∈Mis the center of an evolvingε-neck att= 0, if the solution gij(x,t) in the set {(x,t)| −ε−2Q−1<t≤ 0,d2 t(x,x0)<ε−2Q−1}, whereQ=R(x0,0),is, after scaling with factor Q,ε-close (in theC[ε−1]topology) to the corresponding set of the evolving round cyl inder having scalar curvature one at t= 0. As another consequence of the elliptic type estimate, we hav e the following global structure result obtained by Perelman in [103] for noncompa ct ancientκ-solutions. Corollary 6.4.5. For anyε>0there exists C=C(ε)>0, such that if gij(t) is a nonﬂat ancient κ-solution on a noncompact three-manifold Mfor someκ >0, andMεdenotes the set of points in Mwhich are not centers of evolving ε-necks at t= 0, then att= 0, either the whole manifold Mis the round cylinder S2×Ror its Z2metric quotients, or Mεsatisﬁes the following (i)Mεis compact, (ii) diamMε≤CQ−1 2andC−1Q≤R(x,0)≤CQ, whenever x∈Mε, where Q=R(x0,0)for somex0∈∂Mε. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 395 Proof. We ﬁrst consider the easy case that the curvature operator of the ancient κ-solution has a nontrivial null vector somewhere at some tim e. Let us pull back the solution to its universal cover. By applying Hamilton’s str ong maximum principle and Theorem 6.2.2, we see that the universal cover is the evolvin g round cylinder S2×R. Thus in this case, by the argument in the proof of Lemma 6.4.1, we conclude that the ancient κ-solution is either isometric to the round cylinder S2×Ror one of its Z2 metric quotients (i.e., RP2×R, or the twisted product S2˜×Rwhere Z2ﬂips both S2, orR). We then assume that the curvature operator of the nonﬂat anci entκ-solution is positive everywhere. Firstly we want to show Mεis compact. We argue by contra- diction. Suppose there exists a sequence of points zk, k= 1,2,..., going to inﬁnity (with respect to the metric gij(0)) such that each zkis not the center of any evolving ε-neck. For an arbitrarily ﬁxed point z0∈M, it follows from Theorem 6.4.3(i) that 0<R(z0,0)≤R(zk,0)·ω(R(zk,0)d2 0(zk,z0)) which implies that lim k→∞R(zk,0)d2 0(zk,z0) = +∞. Since the sectional curvature of the ancient κ-solution is positive everywhere, the underlying manifold is diﬀeomorphic to R3, and in particular, orientable. Then as before, by Proposition 6.1.2, Theorem 6.2.2 and Corollary 6 .4.4, we conclude that zk is the center of an evolving ε-neck forksuﬃciently large. This is a contradiction, so we have proved that Mεis compact. Again, we notice that Mis diﬀeomorphic to R3since the curvature operator is positive. According to the resolution of the Schoenﬂies con jecture in three-dimensions, every approximately round two-sphere cross-section throu gh the center of an evolving ε-neck divides Minto two parts such that one of them is diﬀeomorphic to the thr ee-ball B3. Letϕbe the Busemann function on M, it is a standard fact that ϕis convex and proper. Since Mεis compact, Mεis contained in a compact set K=ϕ−1((−∞,A]) for some largeA. We note that each point x∈M\Mεis the center of an ε-neck. It is clear that there is an ε-neckNlying entirely outside K. Consider a point xon one of the boundary components of the ε-neckN. Sincex∈M\Mε, there is an ε-neck adjacent to the initial ε-neck, producing a longer neck. We then take a point on the bou ndary of the second ε-neck and continue. This procedure can either terminate whe n we get intoMεor go on inﬁnitely to produce a semi-inﬁnite (topological) c ylinder. The same procedure can be repeated for the other boundary compon ent of the initial ε-neck. This procedure will give a maximal extended neck ˜N. If˜Nnever touches Mε, the manifold will be diﬀeomorphic to the standard inﬁnite c ylinder, which is a contradiction. If both ends of ˜NtouchMε, then there is a geodesic connecting two points of Mεand passing through N. This is impossible since the function ϕ is convex. So we conclude that one end of ˜Nwill touchMεand the other end will tend to inﬁnity to produce a semi-inﬁnite (topological) cyl inder. Thus we can ﬁnd an approximately round two-sphere cross-section which enclo ses the whole set Mεand touches some point x0∈∂Mε. We next want to show that R(x0,0)1 2·diam(Mε) is bounded from above by some positive constant C=C(ε) depending only on ε. Suppose not; then there exists a sequence of nonﬂat noncompa ct three- dimensional ancient κ-solutions with positive curvature operator such that for t he above chosen points x0∈∂Mεthere would hold (6.4.21) R(x0,0)1 2·diam(Mε)→+∞. 396 H.-D. CAO AND X.-P. ZHU By Proposition 6.4.2, we know that the ancient solutions are κ0-noncollapsed on all scales for some universal positive constant κ0. Let us dilate the ancient solutions around the points x0with the factors R(x0,0). By Corollary 6.4.4, we can extract a convergent subsequence. From the choice of the points x0and (6.4.21), the limit has at least two ends. Then by Toponogov’s splitting theorem the limit is isometric to X×Rfor some nonﬂat two-dimensional ancient κ0-solutionX. SinceMis orientable, we conclude from Theorem 6.2.2 that limit must be the evolvin g round cylinder S2×R. This contradicts the fact that each chosen point x0is not the center of any evolving ε-neck. Therefore we have proved diam(Mε)≤CQ−1 2 for some positive constant C=C(ε) depending only on ε, whereQ=R(x0,0). Finally by combining this diameter estimate with Theorem 6. 4.3(i), we immedi- ately deduce /tildewideC−1Q≤R(x,0)≤/tildewideCQ, wheneverx∈Mε, for some positive constant /tildewideCdepending only on ε. We now can describe the canonical structures for three-dime nsional nonﬂat (com- pact or noncompact) ancient κ-solutions. The following theorem was given by Perel- man in the section 1.5 of [104]. Recently in [34], this canoni cal neighborhood result has been extended to four-dimensional ancient κ-solutions with isotropic curvature pinching. Theorem 6.4.6 ( Canonical neighborhood theorem ).For anyε >0one can ﬁnd positive constants C1=C1(ε)andC2=C2(ε)with the following property. Suppose we have a three-dimensional nonﬂat (compact or noncompact )ancientκ- solution (M,g ij(x,t)). Then either the ancient solution is the round RP2×R, or every point (x,t)has an open neighborhood B, withBt(x,r)⊂B⊂Bt(x,2r)for some 0<r<C 1R(x,t)−1 2, which falls into one of the following three categories: (a)Bis anevolvingε-neck (in the sense that it is the slice at the time tof the parabolic region {(x′,t′)|x′∈B,t′∈[t−ε−2R(x,t)−1,t]}which is, after scaling with factor R(x,t)and shifting the time tto zero,ε-close (in theC[ε−1]topology )to the subset (S2×I)×[−ε−2,0]of the evolving standard round cylinder with scalar curvature 1and length 2ε−1toIat the time zero ), or (b)Bis anevolvingε-cap(in the sense that it is the time slice at the time t of an evolving metric on B3orRP3\¯B3such that the region outside some suitable compact subset of B3orRP3\¯B3is an evolving ε-neck),or (c)Bis a compact manifold (without boundary )with positive sectional curvature (thus it is diﬀeomorphic to the round three-sphere S3or a metric quotient of S3); furthermore, the scalar curvature of the ancient κ-solution on Bat timetis between C−1 2R(x,t)andC2R(x,t), and the volume of Bin case (a)and case (b)satisﬁes (C2R(x,t))−3 2≤Volt(B)≤εr3. Proof. As before, we ﬁrst consider the easy case that the curvature o perator has a nontrivial null vector somewhere at some time. By pulling b ack the solution to THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 397 its universal cover and applying Hamilton’s strong maximum principle and Theorem 6.2.2, we deduce that the universal cover is the evolving rou nd cylinder S2×R. Then exactly as before, by the argument in the proof of Lemma 6.4.1 , we conclude that the ancientκ-solution is isometric to the round S2×R,RP2×R, or the twisted product S2˜×Rwhere Z2ﬂips both S2andR. Clearly each point of the round cylinder S2×R or the twisted product S2˜×Rhas a neighborhood falling into the category (a) or (b) (over RP3\¯B3). We now assume that the curvature operator of the nonﬂat ancie ntκ-solution is positive everywhere. Then the manifold is orientable by the Cheeger-Gromoll theorem [23] for the noncompact case or the Synge theorem [22] for the compact case. Without loss of generality, we may assume εis suitably small, say 0 < ε <1 100. If the nonﬂat ancient κ-solution is noncompact, the conclusions follow immediate ly from the combination of Corollary 6.4.5 and Theorem 6.4.3(i ). Thus we may assume the nonﬂat ancient κ-solution is compact. By Proposition 6.4.2, either the comp act ancientκ-solution is isometric to a metric quotient of the round S3, or it isκ0- noncollapsed on all scales for the universal positive const antκ0. Clearly each point of a metric quotient of the round S3has a neighborhood falling into category (c). Thus we may further assume the ancient κ-solution is also κ0-noncollapsing. We argue by contradiction. Suppose that for some ε∈(0,1 100), there exist a sequence of compact orientable ancient κ0-solutions ( Mk,gk) with positive curvature operator, a sequence of points ( xk,0) withxk∈Mkand sequences of positive constants C1k→ ∞ andC2k=ω(4C2 1k), with the function ωgiven in Theorem 6.4.3, such that for every radius r, 0< r < C 1kR(xk,0)−1 2, any open neighborhood B, with B0(xk,r)⊂B⊂B0(xk,2r), does not fall into one of the three categories (a), (b) and (c), where in the case (a) and case (b), we require the neighbo rhoodBto satisfy the volume estimate (C2kR(xk,0))−3 2≤Vol0(B)≤εr3. By Theorem 6.4.3(i) and the choice of the constants C2kwe see that the diameter of eachMkatt= 0 is at least C1kR(xk,0)−1 2; otherwise we can choose suitable r∈(0,C1kR(xk,0)−1 2) andB=Mk, which falls into the category (c) with the scalar curvature between C−1 2kR(x,0) andC2kR(x,0) onB. Now by scaling the ancient κ0-solutions along the points ( xk,0) with factors R(xk,0), it follows from Corollary 6.4.4 that a sequence of the ancient κ0-solutions converge in the C∞ loctopology to a noncompact orientable ancient κ0-solution. If the curvature operator of the noncompact limit has a nontr ivial null vector somewhere at some time, it follows exactly as before by using the argument in the proof of Lemma 6.4.1 that the orientable limit is isometric t o the round S2×R, or the twisted product S2˜×Rwhere Z2ﬂips both S2andR. Then for klarge enough, a suitable neighborhood B(for suitable r) of the point ( xk,0) would fall into the category (a) or (b) (over RP3\¯B3) with the desired volume estimate. This is a contradiction. If the noncompact limit has positive sectional curvature ev erywhere, then by using Corollary 6.4.5 and Theorem 6.4.3(i) for the noncompact lim it we see that for klarge enough, a suitable neighborhood B(for suitable r) of the point ( xk,0) would fall into category (a) or (b) (over B3) with the desired volume estimate. This is also a contradiction. Finally, the statement on the curvature estimate in the neig hborhoodBfollows directly from Theorem 6.4.3(i). 398 H.-D. CAO AND X.-P. ZHU 7. Ricci Flow on Three-manifolds. We will use the Ricci ﬂow to study the topology of compact orientable three-manifolds. Let Mbe a compact three- dimensional orientable manifold. Arbitrarily given a Riem annian metric on the man- ifold, we evolve it by the Ricci ﬂow. The basic idea is to under stand the topology of the underlying manifold by studying long-time behavior of t he solution of the Ricci ﬂow. We have seen in Chapter 5 that for a compact three-manifo ld with positive Ricci curvature as initial data, the solution to the Ricci ﬂo w tends, up to scalings, to a metric of positive constant curvature. Consequently, a compact three-manifold with positive Ricci curvature is diﬀeomorphic to the round t hree-sphere or a metric quotient of it. However, for general initial metrics, the Ricci ﬂow may deve lop singularities in some parts while it keeps smooth in other parts. Naturally on e would like to cut oﬀ the singularities and continue to run the Ricci ﬂow. If the Ri cci ﬂow still develops singularities after a while, one can do the surgeries and run the Ricci ﬂow again. By repeating this procedure, one will get a kind of “weak” solut ion to the Ricci ﬂow. Furthermore, if the “weak” solution has only a ﬁnite number o f surgeries at any ﬁnite time interval and one can remember what had been cut during th e surgeries, and if the “weak” solution has a well-understood long-time beha vior, then one will also get the topology structure of the initial manifold. This the ory of surgically modiﬁed Ricci ﬂow was ﬁrst developed by Hamilton [64] for compact fou r-manifolds and further developed more recently by Perelman [104] for compact orien table three-manifolds. The main purpose of this chapter is to give a complete and deta iled discussion of the Ricci ﬂow with surgery on three-manifolds. 7.1. Canonical Neighborhood Structures. Let us call a Riemannian metric on a compact orientable three-dimensional manifold normalized if the eigenvalues of its curvature operator at every point are bounded by1 10≥λ≥µ≥ν≥ −1 10, and every geodesic ball of radius one has volume at least one. By t he evolution equation of the curvature and the maximum principle, it is easy to see t hat any solution to the Ricci ﬂow with (compact and three-dimensional) normalized initial metric exists on a maximal time interval [0 ,tmax) withtmax>1. Consider a smooth solution gij(x,t) to the Ricci ﬂow on M×[0,T), whereMis a compact orientable three-manifold and T <+∞. After rescaling, we may always assume the initial metric gij(·,0) is normalized. By Theorem 5.3.2, the solution gij(·,t) then satisﬁes the pinching estimate (7.1.1) R≥(−ν)[log(−ν) + log(1 + t)−3] wheneverν <0 onM×[0,T). Recall the function y=f(x) =x(logx−3),fore2≤x<+∞, is increasing and convex with range −e2≤y <+∞, and its inverse function is also increasing and satisﬁes lim y→+∞f−1(y)/y= 0. We can rewrite the pinching estimate (7.1.1) as (7.1.2) Rm(x,t)≥ −[f−1(R(x,t)(1 +t))/(R(x,t)(1 +t))]R(x,t) onM×[0,T). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 399 Suppose that the solution gij(·,t) becomes singular as t→T. Let us take a sequence of times tk→T, and a sequence of points pk∈Msuch that for some positive constantC,|Rm|(x,t)≤CQkwithQk=|Rm(pk,tk)|for allx∈Mandt∈[0,tk]. Thus, (pk,tk) is a sequence of (almost) maximum points. By applying Hamil ton’s compactness theorem and Perelman’s no local collapsing the orem I as well as the pinching estimate (7.1.2), a sequence of the scalings of the solutiongij(x,t) around the pointspkwith factors Qkconverges to a nonﬂat complete three-dimensional orientab le ancientκ-solution (for some κ >0). For an arbitrarily given ε >0, the canonical neighborhood theorem (Theorem 6.4.6) in the previous chapt er implies that each point in the ancient κ-solution has a neighborhood which is either an evolving ε-neck, or an evolving ε-cap, or a compact (without boundary) positively curved man ifold. This gives the structure of singularities coming from a sequ ence of (almost) maximum points. However the above argument does not work for singularities c oming from a se- quence of points ( yk,τk) withτk→Tand|Rm(yk,τk)| →+∞when|Rm(yk,τk)|is not comparable with the maximum of the curvature at the time τk, since we cannot take a limit directly. In [103], Perelman developed a reﬁned rescaling argument to obtain the following singularity structure theorem. We rem ark that our statement of the singularity structure theorem below is slightly diﬀere nt from Perelman’s original statement (cf. Theorem 12.1 of [103]). While Perelman assum ed the condition of κ-noncollapsing on scales less than r0, we assume that the initial metric is normalized so that from the rescaling argument one can get the κ-noncollapsing on all scales for the limit solutions. Theorem 7.1.1 ( Singularity structure theorem ).Givenε>0andT0>1, one can ﬁndr0>0with the following property. If gij(x,t),x∈Mandt∈[0,T)with 1< T≤T0, is a solution to the Ricci ﬂow on a compact orientable three- manifold Mwith normalized initial metric, then for any point (x0,t0)witht0≥1andQ= R(x0,t0)≥r−2 0, the solution in {(x,t)|d2 t0(x,x0)<ε−2Q−1, t0−ε−2Q−1≤t≤t0} is, after scaling by the factor Q,ε-close (in theC[ε−1]-topology )to the corresponding subset of some orientable ancient κ-solution (for someκ>0). Proof. Since the initial metric is normalized, it follows from the n o local collapsing theorem I or I’ (and their proofs) that there is a positive con stantκ, depending only onT0, such that the solution in Theorem 7.1.1 is κ-noncollapsed on all scales less than√T0. LetC(ε) be a positive constant larger than or equal to ε−2. It suﬃces to prove that there exists r0>0 such that for any point ( x0,t0) witht0≥1 andQ= R(x0,t0)≥r−2 0, the solution in the parabolic region {(x,t)∈M×[0,T)|d2 t0(x,x0)< C(ε)Q−1,t0−C(ε)Q−1≤t≤t0}is, after scaling by the factor Q,ε-close to the corresponding subset of some orientable ancient κ-solution. The constant C(ε) will be determined later. We argue by contradiction. Suppose for some ε >0, there exist a sequence of solutions (Mk,gk(·,t)) to the Ricci ﬂow on compact orientable three-manifolds wi th normalized initial metrics, deﬁned on the time intervals [0 ,Tk) with 1< T k≤T0, a sequence of positive numbers rk→0, and a sequence of points xk∈Mkand timestk≥1 withQk=Rk(xk,tk)≥r−2 ksuch that each solution ( Mk,gk(·,t)) in the parabolic region {(x,t)∈Mk×[0,Tk)|d2 tk(x,xk)<C(ε)Q−1 k,tk−C(ε)Q−1 k≤t≤tk} is not, after scaling by the factor Qk,ε-close to the corresponding subset of any orientable ancient κ-solution, where Rkdenotes the scalar curvature of ( Mk,gk). For each solution ( Mk,gk(·,t)), we may adjust the point ( xk,tk) withtk≥1 2 400 H.-D. CAO AND X.-P. ZHU and withQk=Rk(xk,tk) to be as large as possible so that the conclusion of the theorem fails at ( xk,tk), but holds for any ( x,t)∈Mk×[tk−HkQ−1 k,tk] satisfying Rk(x,t)≥2Qk, whereHk=1 4r−2 k→+∞ask→+∞.Indeed, suppose not, by setting (xk1,tk1) = (xk,tk), we can choose a sequence of points ( xkl,tkl)∈Mk× [tk(l−1)−HkRk(xk(l−1),tk(l−1))−1,tk(l−1)] such that Rk(xkl,tkl)≥2Rk(xk(l−1),tk(l−1)) and the conclusion of the theorem fails at ( xkl,tkl) for eachl= 2,3,....Since the solution is smooth, but Rk(xkl,tkl)≥2Rk(xk(l−1),tk(l−1))≥ ··· ≥ 2l−1Rk(xk,tk), and tkl≥tk(l−1)−HkRk(xk(l−1),tk(l−1))−1 ≥tk−Hkl−1/summationdisplay i=11 2i−1Rk(xk,tk)−1 ≥1 2, this process must terminate after a ﬁnite number of steps and the last element ﬁts. Let (Mk,˜gk(·,t), xk) be the rescaled solutions obtained by rescaling (Mk,gk(·,t)) aroundxkwith the factors Qk=Rk(xk,tk) and shifting the time tk to the new time zero. Denote by ˜Rkthe rescaled scalar curvature. We will show that a subsequence of the orientable rescaled solutions ( Mk,˜gk(·,t),xk) converges in the C∞ loctopology to an orientable ancient κ-solution, which is a contradiction. In the following we divide the argument into four steps. Step1. First of all, we need a local bound on curvatures. Lemma 7.1.2. For each (¯x,¯t)withtk−1 2HkQ−1 k≤¯t≤tk, we haveRk(x,t)≤4¯Qk whenever ¯t−c¯Q−1 k≤t≤¯tandd2¯t(x,¯x)≤c¯Q−1 k, where ¯Qk=Qk+Rk(¯x,¯t)andc>0 is a small universal constant. Proof. Consider any point ( x,t)∈B¯t(¯x,(c¯Q−1 k)1 2)×[¯t−c¯Q−1 k,¯t] withc>0 to be determined. If Rk(x,t)≤2Qk, there is nothing to show. If Rk(x,t)>2Qk, consider a space-time curve γfrom (x,t) to (¯x,¯t) that goes straight from ( x,t) to (x,¯t) and goes from (x,¯t) to (¯x,¯t) along a minimizing geodesic (with respect to the metric gk(·,¯t)). If there is a point on γwith the scalar curvature 2 Qk, lety0be the nearest such point to (x,t). If not, put y0= (¯x,¯t).On the segment of γfrom (x,t) toy0, the scalar curvature is at least 2 Qk. According to the choice of the point ( xk,tk), the solution along the segment is ε-close to that of some ancient κ-solution. It follows from Theorem 6.4.3 (ii) that |∇(R−1 2 k)| ≤2ηand/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ ∂t(R−1 k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2η on the segment. (Here, without loss of generality, we may ass umeεis suitably small). Then by choosing c >0 (depending only on η) small enough we get the desired curvature bound by integrating the above derivative estima tes along the segment. This proves the lemma. Step2. Next we want to show that for each A <+∞, there exist a positive constantC(A) (independent of k) such that the curvatures of the rescaled solutions THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 401 ˜gk(·,t) at the new time t= 0 (corresponding to the original times tk) satisfy the estimate |/tildewidestRmk|(y,0)≤C(A) wheneverd˜gk(·,0)(y,xk)≤Aandk≥1. For eachρ≥0, set M(ρ) = sup {˜Rk(x,0)|k≥1,x∈Mkwithd0(x,xk)≤ρ} and ρ0= sup{ρ≥0|M(ρ)<+∞}. By the pinching estimate (7.1.1), it suﬃces to show ρ0= +∞. Note thatρ0>0 by applying Lemma 7.1.2 with (¯ x,¯t) = (xk,tk).We now argue by contradiction to show ρ0= +∞.Suppose not, we may ﬁnd (after passing to a subsequence if necessary) a sequence of points yk∈Mkwithd0(xk,yk)→ρ0<+∞ and˜Rk(yk,0)→+∞. Letγk(⊂Mk) be a minimizing geodesic segment from xkto yk. Letzk∈γkbe the point on γkclosest toykwith˜Rk(zk,0) = 2, and let βkbe the subsegment of γkrunning from zktoyk. By Lemma 7.1.2 the length of βkis bounded away from zero independent of k. By the pinching estimate (7.1.1), for each ρ<ρ 0, we have a uniform bound on the curvatures on the open balls B0(xk,ρ)⊂(Mk,˜gk). The injectivity radii of the rescaled solutions ˜ gkat the points xkand the time t= 0 are also uniformly bounded from below by the κ-noncollapsing property. Therefore by Lemma 7.1.2 and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked sequence ( B0(xk,ρ0),˜gk(·,0),xk) converges in the C∞ loctopology to a marked (noncomplete) manifold ( B∞,˜g∞,x∞), the segments γkconverge to a geodesic segment (missing an endpoint) γ∞⊂B∞ emanating from x∞, andβkconverges to a subsegment β∞ofγ∞. Let ¯B∞denote the completion of ( B∞,˜g∞), andy∞∈¯B∞the limit point of γ∞. Denote by ˜R∞the scalar curvature of ( B∞,˜g∞). Since the rescaled scalar curva- tures ˜Rkalongβkare at least 2, it follows from the choice of the points ( xk,0) that for anyq0∈β∞, the manifold ( B∞,˜g∞) in{q∈B∞|dist2 ˜g∞(q,q0)<C(ε)(˜R∞(q0))−1} is 2ε-close to the corresponding subset of (a time slice of) some o rientable ancient κ-solution. Then by Theorem 6.4.6, we know that the orientabl e ancientκ-solution at each point ( x,t) has a radius r, 0< r < C 1(2ε)R(x,t)−1 2, such that its canon- ical neighborhood B, withBt(x,r)⊂B⊂Bt(x,2r), is either an evolving 2 ε-neck, or an evolving 2 ε-cap, or a compact manifold (without boundary) diﬀeomorphi c to a metric quotient of the round three-sphere S3, and moreover the scalar curvature is be- tween (C2(2ε))−1R(x,t) andC2(2ε)R(x,t), whereC1(2ε) andC2(2ε) are the positive constants in Theorem 6.4.6. We now choose C(ε) = max {2C2 1(2ε),ε−2}. By the local curvature estimate in Lemma 7.1.2, we see that the scalar curvature ˜R∞becomes unbounded when approaching y∞alongγ∞. This implies that the canonical neighborhood around q0 cannot be a compact manifold (without boundary) diﬀeomorph ic to a metric quotient of the round three-sphere S3. Note that γ∞is shortest since it is the limit of a sequence of shortest geodesics. Without loss of generality, we may as sumeεis suitably small (say,ε≤1 100). These imply that as q0gets suﬃciently close to y∞, the canonical neighborhood around q0cannot be an evolving 2 ε-cap. Thus we conclude that each q0∈γ∞suﬃciently close to y∞is the center of an evolving 2 ε-neck. 402 H.-D. CAO AND X.-P. ZHU Let U=/uniondisplay q0∈γ∞B(q0,4π(˜R∞(q0))−1 2) (⊂(B∞,˜g∞)), whereB(q0,4π(˜R∞(q0))−1 2) is the ball centered at q0∈B∞with the radius 4π(˜R∞(q0))−1 2. ClearlyUhas nonnegative sectional curvature by the pinching es- timate (7.1.1). Since the metric ˜ g∞is cylindrical at any point q0∈γ∞which is suﬃciently close to y∞, we see that the metric space U=U∪ {y∞}by adding in the pointy∞, is locally complete and intrinsic near y∞. Furthermore y∞cannot be an interior point of any geodesic segment in U. This implies the curvature of Uat y∞is nonnegative in the Alexandrov sense. It is a basic result i n Alexandrov space theory (see for example Theorem 10.9.3 and Corollary 10.9.5 of [9]) that there exists a three-dimensional tangent cone Cy∞Uaty∞which is a metric cone. It is clear that its aperture is ≤10ε, thus the tangent cone is nonﬂat. Pick a point p∈Cy∞Usuch that the distance from the vertex y∞topis one and it is nonﬂat around p. Then the ball B(p,1 2)⊂Cy∞Uis the Gromov-Hausdorﬀ limit of the scalings of a sequence of balls B0(pk,sk)⊂(Mk,˜gk(·,0)) by some factors ak, wheresk→0+. Since the tangent cone is three-dimensional and nonﬂat aro und p, the factors akmust be comparable with ˜Rk(pk,0). By using the local curvature estimate in Lemma 7.1.2, we actually have the convergence in theC∞ loctopology for the solutions ˜ gk(·,t) on the balls B0(pk,sk) and over some time interval t∈[−δ,0] for some suﬃciently small δ >0. The limiting ball B(p,1 2)⊂Cy∞Uis a piece of the nonnegative curved and nonﬂat metric cone whose radial d irections are all Ricci ﬂat. On the other hand, by applying Hamilton’s strong maximu m principle to the evolution equation of the Ricci curvature tensor as in the pr oof of Lemma 6.3.1, the limiting ball B(p,1 2) would split oﬀ all radial directions isometrically (and lo cally). Since the limit is nonﬂat around p, this is impossible. Therefore we have proved that the curvatures of the rescaled solutions ˜ gk(·,t) at the new times t= 0 (corresponding to the original times tk) stay uniformly bounded at bounded distances from xkfor all k. We have proved that for each A <+∞, the curvature of the marked manifold (Mk,˜gk(·,0),xk) at each point y∈Mkwith distance from xkat mostAis bounded byC(A). Lemma 7.1.2 extends this curvature control to a backward p arabolic neigh- borhood centered at ywhose radius depends only on the distance from ytoxk. Thus by Shi’s local derivative estimates (Theorem 1.4.2) we can control all deriva- tives of the curvature in such backward parabolic neighborh oods. Then by using the κ-noncollapsing and Hamilton’s compactness theorem (Theor em 4.1.5), we can take aC∞ locsubsequent limit to obtain ( M∞,˜g∞(·,t),x∞), which is κ-noncollapsed on all scales and is deﬁned on a space-time open subset of M∞×(−∞,0] containing the time sliceM∞× {0}. Clearly it follows from the pinching estimate (7.1.1) that the limit (M∞,˜g∞(·,0),x∞) has nonnegative curvature operator (and hence nonnegativ e sectional curvature). Step3. We further claim that the limit ( M∞,˜g∞(·,0),x∞) at the time slice {t= 0}has bounded curvature. We know that the sectional curvature of the limit ( M∞,˜g∞(·,0),x∞) is nonnega- tive everywhere. Argue by contradiction. Suppose the curva ture of (M∞,˜g∞(·,0),x∞) is not bounded, then by Lemma 6.1.4, there exists a sequence o f pointsqj∈M∞di- verging to inﬁnity such that their scalar curvatures ˜R∞(qj,0)→+∞asj→+∞ THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 403 and ˜R∞(x,0)≤4˜R∞(qj,0) forx∈B(qj,j//radicalig ˜R∞(qj,0))⊂(M∞,˜g∞(·,0)). By combining with Lemma 7.1.2 and theκ-noncollapsing, a subsequence of the rescaled and marked ma nifolds (M∞,˜R∞(qj,0)˜g∞(·,0),qj) converges in the C∞ loctopology to a smooth nonﬂat limit Y. By Proposition 6.1.2, the new limit Yis isometric to a metric product N×Rfor some two-dimensional manifold N. On the other hand, in view of the choice of the points (xk,tk), the original limit ( M∞,˜g∞(·,0),x∞) at the point qjhas a canonical neighborhood which is either a 2 ε-neck, a 2ε-cap, or a compact manifold (without boundary) diﬀeomorphic to a metric quotient of the round S3. It follows that for j large enough, qjis the center of a 2 ε-neck of radius ( ˜R∞(qj,0))−1 2. Without loss of generality, we may further assume that 2 ε < ε 0, whereε0is the positive constant given in Proposition 6.1.1. Since ( ˜R∞(qj,0))−1 2→0 asj→+∞, this contradicts Proposition 6.1.1. So the curvature of ( M∞,˜g∞(·,0)) is bounded. Step4. Finally we want to extend the limit backwards in time to −∞. By Lemma 7.1.2 again, we now know that the limiting solution ( M∞,˜g∞(·,t)) is deﬁned on a backward time interval [ −a,0] for some a>0. Denote by t′= inf{˜t|we can take a smooth limit on ( ˜t,0] (with bounded curvature at each time slice) from a subsequence of the convergent rescaled solutions ˜ gk}. We ﬁrst claim that there is a subsequence of the rescaled solu tions ˜gkwhich converges in theC∞ loctopology to a smooth limit ( M∞,˜g∞(·,t)) on the maximal time interval (t′,0]. Indeed, lett′ kbe a sequence of negative numbers such that t′ k→t′and there exist smooth limits ( M∞,˜gk ∞(·,t)) deﬁned on ( t′ k,0]. For each k, the limit has nonnegative sectional curvature and has bounded curvature at each time s lice. Moreover by Lemma 7.1.2, the limit has bounded curvature on each subinterval [ −b,0]⊂(t′ k,0]. Denote by˜Qthe scalar curvature upper bound of the limit at time zero (wh ere˜Qis the same for allk). Then we can apply the Li-Yau-Hamilton estimate (Corollar y 2.5.7) to get ˜Rk ∞(x,t)≤˜Q/parenleftbigg−t′ k t−t′ k/parenrightbigg , where ˜Rk ∞(x,t) are the scalar curvatures of the limits ( M∞,˜gk ∞(·,t)). Hence by the deﬁnition of convergence and the above curvature estimates , we can ﬁnd a subsequence of the above convergent rescaled solutions ˜ gkwhich converges in the C∞ loctopology to a smooth limit ( M∞,˜g∞(·,t)) on the maximal time interval ( t′,0]. We next claim that t′=−∞. Suppose not, then by Lemma 7.1.2, the curvature of the limit ( M∞,˜g∞(·,t)) becomes unbounded as t→t′>−∞. By applying the maximum principle to the evolution equation of the scalar curvature, we see that the i nﬁmum of the scalar curvature is nondecreasing in time. Note that ˜R∞(x∞,0) = 1. Thus there exists some point y∞∈M∞such that ˜R∞/parenleftig y∞,t′+c 10/parenrightig <3 2 404 H.-D. CAO AND X.-P. ZHU wherec >0 is the universal constant in Lemma 7.1.2. By using Lemma 7.1 .2 again we see that the limit ( M∞,˜g∞(·,t)) in a small neighborhood of the point ( y∞,t′+c 10) extends backwards to the time interval [ t′−c 10,t′+c 10]. We remark that the distances at timetand time 0 are roughly equivalent in the following sense (7.1.3) dt(x,y)≥d0(x,y)≥dt(x,y)−const. for anyx,y∈M∞andt∈(t′,0]. Indeed from the Li-Yau-Hamilton inequality (Corollary 2.5.7) we have the estimate ˜R∞(x,t)≤˜Q/parenleftbigg−t′ t−t′/parenrightbigg ,onM∞×(t′,0]. By applying Lemma 3.4.1 (ii), we have dt(x,y)≤d0(x,y) + 30( −t′)/radicalig ˜Q for anyx,y∈M∞andt∈(t′,0]. On the other hand, since the curvature of the limit metric ˜g∞(·,t) is nonnegative, we have dt(x,y)≥d0(x,y) for anyx,y∈M∞andt∈(t′,0]. Thus we obtain the estimate (7.1.3). Let us still denote by ( Mk,˜gk(·,t)) the subsequence which converges on the maxi- mal time interval ( t′,0]. Consider the rescaled sequence ( Mk,˜gk(·,t)) with the marked pointsxkreplaced by the associated sequence of points yk→y∞and the (original unshifted) times tkreplaced by any sk∈[tk+ (t′−c 20)Q−1 k,tk+ (t′+c 20)Q−1 k]. It follows from Lemma 7.1.2 that for klarge enough, the rescaled solutions ( Mk,˜gk(·,t)) atyksatisfy ˜Rk(yk,t)≤10 for allt∈[t′−c 10,t′+c 10]. By applying the same arguments as in the above Step 2, we conclude that for any A>0, there is a positive constant C(A)<+∞such that ˜Rk(x,t)≤C(A) for all (x,t) withdt(x,yk)≤Aandt∈[t′−c 20,t′+c 20]. The estimate (7.1.3) implies that there is a positive constant A0such that for arbitrarily given small ǫ′∈(0,c 100), forklarge enough, there hold dt(xk,yk)≤A0 for allt∈[t′+ǫ′,0]. By combining with Lemma 7.1.2, we then conclude that for a ny A >0, there is a positive constant ˜C(A) such that for klarge enough, the rescaled solutions (Mk,˜gk(·,t)) satisfy ˜Rk(x,t)≤˜C(A) for allx∈˜B0(xk,A) andt∈[t′−c 100(C(A))−1,0]. Now, by taking convergent subsequences from the (original) rescaled solutions (Mk,˜gk(·,t),xk), we see that the limiting solution ( M∞,˜g∞(·,t)) is deﬁned on a space- time open subset of M∞×(−∞,0] containing M∞×[t′,0]. By repeating the argument THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 405 of Step 3 and using Lemma 7.1.2, we further conclude the limit (M∞,˜g∞(·,t)) has uniformly bounded curvature on M∞×[t′,0]. This is a contradiction. Therefore we have proved a subsequence of the rescaled solut ions (Mk,˜gk(·,t),xk) converges to an orientable ancient κ-solution, which gives the desired contradiction. This completes the proof of the theorem. We remark that this singularity structure theorem has been e xtended by Chen and the second author in [34] to the Ricci ﬂow on compact four- manifolds with positive isotropic curvature. 7.2. Curvature Estimates for Smooth Solutions. Let us consider solutions to the Ricci ﬂow on compact orientable three-manifolds with normalized initial met- rics. The above singularity structure theorem (Theorem 7.1 .1) tells us that the so- lutions around high curvature points are suﬃciently close t o ancientκ-solutions. It is thus reasonable to expect that the elliptic type estimate (Theorem 6.4.3) and the curvature estimate via volume growth (Theorem 6.3.3) for an cientκ-solutions are heritable to general solutions of the Ricci ﬂow on three-man ifolds. The main purpose of this section is to establish such curvature estimates. In the ﬁfth section of this chapter, we will further extend these estimates to surgical ly modiﬁed solutions. The ﬁrst result of this section is an extension of the ellipti c type estimate (The- orem 6.4.3). This result is reminiscent of the second step in the proof of Theorem 7.1.1. Theorem 7.2.1 ( Perelman [103] ).For anyA <+∞, there exist K=K(A)< +∞andα=α(A)>0with the following property. Suppose we have a solution to the Ricci ﬂow on a three-dimensional, compact and orienta ble manifold Mwith normalized initial metric. Suppose that for some x0∈Mand somer0>0with r0<α, the solution is deﬁned for 0≤t≤r2 0and satisﬁes |Rm|(x,t)≤r−2 0,for0≤t≤r2 0, d0(x,x0)≤r0, and Vol0(B0(x0,r0))≥A−1r3 0. ThenR(x,r2 0)≤Kr−2 0wheneverdr2 0(x,x0)<Ar 0. Proof. Given any large A>0 and letting α>0 be chosen later, by Perelman’s no local collapsing theorem II (Theorem 3.4.2), there exists a positive constant κ=κ(A) (independent of α) such that any complete solution satisfying the assumption s of the theorem is κ-noncollapsed on scales ≤r0over the region {(x,t)|1 5r2 0≤t≤ r2 0, dt(x,x0)≤5Ar0}.Set ε= min/braceleftbigg1 4ε0,1 100/bracerightbigg , whereε0is the positive constant in Proposition 6.1.1. We ﬁrst prove the following assertion. Claim. For the above ﬁxed ε >0, one can ﬁnd K=K(A,ε)<+∞such that if we have a three-dimensional complete orientable solutio n with normalized initial metric and satisfying |Rm|(x,t)≤r−2 0for 0≤t≤r2 0, d0(x,x0)≤r0, 406 H.-D. CAO AND X.-P. ZHU and Vol0(B0(x0,r0))≥A−1r3 0 for somex0∈Mand somer0>0, then for any point x∈Mwithdr2 0(x,x0)<3Ar0, either R(x,r2 0)<Kr−2 0 or the subset {(y,t)|d2 r2 0(y,x)≤ε−2R(x,r2 0)−1, r2 0−ε−2R(x,r2 0)−1≤t≤r2 0}around the point ( x,r2 0) isε-close to the corresponding subset of an orientable ancient κ- solution. Notice that in this assertion we don’t impose the restrictio n ofr0<α, so we can consider for the moment r0>0 to be arbitrary in proving the above claim. Note that the assumption on the normalization of the initial metr ic is just to ensure the pinching estimate. By scaling, we may assume r0= 1. The proof of the claim is essentially adapted from that of Theorem 7.1.1. But we will m eet the diﬃculties of adjusting points and verifying a local curvature estimate. Suppose that the claim is not true. Then there exist a sequenc e of solutions (Mk,gk(·,t)) to the Ricci ﬂow satisfying the assumptions of the claim wi th the origins x0k, and a sequence of positive numbers Kk→ ∞, timestk= 1 and points xk∈ Mkwithdtk(xk,x0k)<3Asuch thatQk=Rk(xk,tk)≥Kkand the solution in {(x,t)|tk−C(ε)Q−1 k≤t≤tk, d2 tk(x,xk)≤C(ε)Q−1 k}is not, after scaling by the factorQk,ε-close to the corresponding subset of any orientable ancien tκ-solution, whereRkdenotes the scalar curvature of ( Mk,gk(·,t)) andC(ε)(≥ε−2) is the constant deﬁned in the proof of Theorem 7.1.1. As before we need to ﬁrst adjust the point (xk,tk) withtk≥1 2anddtk(xk,x0k)<4Aso thatQk=Rk(xk,tk)≥Kkand the conclusion of the claim fails at ( xk,tk), but holds for any ( x,t) satisfying Rk(x,t)≥ 2Qk, tk−HkQ−1 k≤t≤tkanddt(x,x0k)< dtk(xk,x0k) +H1 2 kQ−1 2 k, whereHk= 1 4Kk→ ∞,ask→+∞. Indeed, by starting with ( xk1,tk1) = (xk,1) we can choose ( xk2,tk2)∈Mk× (0,1] withtk1−HkRk(xk1,tk1)−1≤tk2≤tk1, anddtk2(xk2,x0k)< dtk1(xk1,x0k) + H1 2 kRk(xk1,tk1)−1 2such thatRk(xk2,tk2)≥2Rk(xk1,tk1) and the conclusion of the claim fails at ( xk2,tk2); otherwise we have the desired point. Repeating this proce ss, we can choose points ( xki,tki),i= 2,...,j , such that Rk(xki,tki)≥2Rk(xki−1,tki−1), tki−1−HkRk(xki−1,tki−1)−1≤tki≤tki−1, dtki(xki,x0k)<dtki−1(xki−1,x0k) +H1 2 kRk(xki−1,tki−1)−1 2, and the conclusion of the claim fails at the points ( xki,tki),i= 2,...,j . These inequalities imply Rk(xkj,tkj)≥2j−1Rk(xk1,tk1)≥2j−1Kk, 1≥tkj≥tk1−Hkj−2/summationdisplay i=01 2iRk(xk1,tk1)−1≥1 2, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 407 and dtkj(xkj,x0k)<dtk1(xk1,x0k) +H1 2 kj−2/summationdisplay i=01 (√ 2)iRk(xk1,tk1)−1 2<4A. Since the solutions are smooth, this process must terminate after a ﬁnite number of steps to give the desired point, still denoted by ( xk,tk). For each adjusted ( xk,tk), let [t′,tk] be the maximal subinterval of [ tk− 1 2ε−2Q−1 k,tk] so that the conclusion of the claim with K= 2Qkholds on P/parenleftbigg xk,tk,1 10H1 2 kQ−1 2 k,t′−tk/parenrightbigg =/braceleftbigg (x,t)|x∈Bt/parenleftbigg xk,1 10H1 2 kQ−1 2 k/parenrightbigg ,t∈[t′,tk]/bracerightbigg for all suﬃciently large k. We now want to show t′=tk−1 2ε−2Q−1 k. Consider the scalar curvature Rkat the point xkover the time interval [ t′,tk]. If there is a time ˜t∈[t′,tk] satisfying Rk(xk,˜t)≥2Qk, we let ˜tbe the ﬁrst of such time from tk. Then the solution ( Mk,gk(·,t)) around the point xkover the time interval [ ˜t−1 2ε−2Q−1 k,˜t] isε-close to some orientable ancient κ-solution. Note from the Li-Yau-Hamilton inequality that the scalar curvature o f any ancient κ-solution is pointwise nondecreasing in time. Consequently, we have t he following curvature estimate Rk(xk,t)≤2(1 +ε)Qk fort∈[˜t−1 2ε−2Q−1 k,tk] (ort∈[t′,tk] if there is no such time ˜t). By combining with the elliptic type estimate for ancient κ-solutions (Theorem 6.4.3) and the Hamilton- Ivey pinching estimate, we further have (7.2.1) |Rm(x,t)| ≤5ω(1)Qk for allx∈Bt(xk,(3Qk)−1 2) andt∈[˜t−1 2ε−2Q−1 k,tk] (ort∈[t′,tk]) and all suﬃciently largek, whereωis the positive function in Theorem 6.4.3. For any point ( x,t) with ˜t−1 2ε−2Q−1 k≤t≤tk(ort∈[t′,tk]) anddt(x,xk)≤ 1 10H1 2 kQ−1 2 k, we divide the discussion into two cases. Case(1):dt(xk,x0k)≤3 10H1 2 kQ−1 2 k. dt(x,x0k)≤dt(x,xk) +dt(xk,x0k) (7.2.2) ≤1 10H1 2 kQ−1 2 k+3 10H1 2 kQ−1 2 k ≤1 2H1 2 kQ−1 2 k. Case(2):dt(xk,x0k)>3 10H1 2 kQ−1 2 k. From the curvature bound (7 .2.1) and the assumption, we apply Lemma 3.4.1(ii) withr0=Q−1 2 kto get d dt(dt(xk,x0k))≥ −20(ω(1) + 1)Q1 2 k, 408 H.-D. CAO AND X.-P. ZHU and then for klarge enough, dt(xk,x0k)≤dˆt(xk,x0k) + 20(ω(1) + 1)ε−2Q−1 2 k ≤dˆt(xk,x0k) +1 10H1 2 kQ−1 2 k, where ˆt∈(t,tk] satisﬁes the property that ds(xk,x0k)≥3 10H1 2 kQ−1 2 kwhenevers∈[t,ˆt]. So we have dt(x,x0k)≤dt(x,xk) +dt(xk,x0k) (7.2.3) ≤1 10H1 2 kQ−1 2 k+dˆt(xk,x0k) +1 10H1 2 kQ−1 2 k ≤dtk(xk,x0k) +1 2H1 2 kQ−1 2 k, for all suﬃciently large k. Then the combination of (7.2.2), (7.2.3) and the choice of the points ( xk,tk) impliest′=tk−1 2ε−2Q−1 kfor all suﬃciently large k. (Here we also used the maximality of the subinterval [ t′,tk] in the case that there is no time in [t′,tk] withRk(xk,·)≥2Qk.) Now we rescale the solutions ( Mk,gk(·,t)) into (Mk,˜gk(·,t)) around the points xk by the factors Qk=Rk(xk,tk) and shift the times tkto the new times zero. Then the same arguments from Step 1 to Step 3 in the proof of Theorem 7.1.1 prove that a subsequence of the rescaled solutions ( Mk,˜gk(·,t)) converges in the C∞ loctopology to a limiting (complete) solution ( M∞,˜g∞(·,t)), which is deﬁned on a backward time interval [ −a,0] for some a >0. (The only modiﬁcation is in Lemma 7.1.2 of Step 1 by further requiring tk−1 4ε−2Q−1 k≤¯t≤tk). We next study how to adapt the argument of Step 4 in the proof of Theorem 7.1.1. As before, we have a maximal time interval ( t∞,0] for which we can take a smooth limit (M∞,˜g∞(·,t),x∞) from a subsequence of the rescaled solutions ( Mk,˜gk(·,t),xk). We want to show t∞=−∞. Suppose not; then t∞>−∞.Letc>0 be a positive constant much smaller than 1 10ε−2. Note that the inﬁmum of the scalar curvature is nondecreasi ng in time. Then we can ﬁnd some point y∞∈M∞and some time t=t∞+θwith 0< θ <c 3such that˜R∞(y∞,t∞+θ)≤3 2. Consider the (unrescaled) scalar curvature Rkof (Mk,gk(·,t)) at the point xk over the time interval [ tk+ (t∞+θ 2)Q−1 k,tk]. Since the scalar curvature ˜R∞of the limit onM∞×[t∞+θ 3,0] is uniformly bounded by some positive constant C, we have the curvature estimate Rk(xk,t)≤2CQk for allt∈[tk+ (t∞+θ 2)Q−1 k,tk] and all suﬃciently large k. Then by repeat- ing the same arguments as in deriving (7.2.1), (7.2.2) and (7 .2.3), we deduce that the conclusion of the claim with K= 2Qkholds on the parabolic neighborhood P(xk,tk,1 10H1 2 kQ−1 2 k,(t∞+θ 2)Q−1 k) for all suﬃciently large k. Let (yk,tk+ (t∞+θk)Q−1 k) be a sequence of associated points and times in the (unrescaled) solutions ( Mk,gk(·,t)) so that after rescaling, the sequence converges to the (y∞,t∞+θ) in the limit. Clearlyθ 2≤θk≤2θfor all suﬃciently large k. Then, by considering the scalar curvature Rkat the point ykover the time interval [tk+(t∞−c 3)Q−1 k,tk+(t∞+θk)Q−1 k], the above argument (as in deriving the similar THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 409 estimates (7.2.1)-(7.2.3)) implies that the conclusion of the claim with K= 2Qkholds on the parabolic neighborhood P(yk,tk,1 10H1 2 kQ−1 2 k,(t∞−c 3)Q−1 k) for all suﬃciently largek. In particular, we have the curvature estimate Rk(yk,t)≤4(1 +ε)Qk fort∈[tk+ (t∞−c 3)Q−1 k,tk+ (t∞+θk)Q−1 k] for all suﬃciently large k. We now consider the rescaled sequence ( Mk,˜gk(·,t)) with the marked points re- placed byykand the times replaced by sk∈[tk+ (t∞−c 4)Q−1 k,tk+ (t∞+c 4)Q−1 k]. By applying the same arguments from Step 1 to Step 3 in the proo f of Theorem 7.1.1 and the Li-Yau-Hamilton inequality as in Step 4 of Theorem 7. 1.1, we conclude that there is some small constant a′>0 such that the original limit ( M∞,˜g∞(·,t)) is ac- tually well deﬁned on M∞×[t∞−a′,0] with uniformly bounded curvature. This is a contradiction. Therefore we have checked the claim. To ﬁnish the proof, we next argue by contradiction. Suppose t here exist sequences of positive numbers Kk→+∞,αk→0, ask→+∞, and a sequence of solutions (Mk,gk(·,t)) to the Ricci ﬂow satisfying the assumptions of the theorem with origins x0kand with radii r0ksatisfyingr0k< α ksuch that for some points xk∈Mkwith dr2 0k(xk,x0k)<Ar 0kwe have (7.2.4) R(xk,r2 0k)>K kr−2 0k for allk. Let (Mk,ˆgk(·,t),x0k) be the rescaled solutions of ( Mk,gk(·,t)) around the originsx0kby the factors r−2 0kand shifting the times r2 0kto the new times zero. The above claim tells us that for klarge, any point ( y,0)∈(Mk,ˆgk(·,0),x0k) with dˆgk(·,0)(y,x0k)<3Aand with the rescaled scalar curvature ˆRk(y,0)> K khas a canonical neighborhood which is either a 2 ε-neck, or a 2 ε-cap, or a compact manifold (without boundary) diﬀeomorphic to a metric quotient of the round three-sphere. Note that the pinching estimate (7.1.1) and the condition αk→0 imply any sub- sequential limit of the rescaled solutions ( Mk,ˆgk(·,t),x0k) must have nonnegative sectional curvature. Thus the same argument as in Step 2 of th e proof of Theorem 7.1.1 shows that for all suﬃciently large k, the curvatures of the rescaled solutions at the time zero stay uniformly bounded at those points whose distances from the originsx0kdo not exceed 2 A. This contradicts (7.2.4) for klarge enough. Therefore we have completed the proof of the theorem. The next result is a generalization of the curvature estimat e via volume growth in Theorem 6.3.3 (ii) where the condition on the curvature lo wer bound over a time interval is replaced by that at a time slice only. Theorem 7.2.2 ( Perelman [103] ).For anyw >0there exist τ=τ(w)>0, K=K(w)<+∞,α=α(w)>0with the following property. Suppose we have a three-dimensional, compact and orientable solution to th e Ricci ﬂow deﬁned on M×[0,T)with normalized initial metric. Suppose that for some radiu sr0>0with r0<αand a point (x0,t0)∈M×[0,T)withT >t 0≥4τr2 0, the solution on the ball Bt0(x0,r0)satisﬁes Rm(x,t0)≥ −r−2 0on B t0(x0,r0), and Volt0(Bt0(x0,r0))≥wr3 0. 410 H.-D. CAO AND X.-P. ZHU ThenR(x,t)≤Kr−2 0whenevert∈[t0−τr2 0,t0]anddt(x,x0)≤1 4r0. Proof. If we knew that Rm(x,t)≥ −r−2 0 for allt∈[0,t0] anddt(x,x0)≤r0, then we could just apply Theorem 6.3.3 (ii) and takeτ(w) =τ0(w)/2,K(w) =C(w) + 2B(w)/τ0(w). Now ﬁx these values of τand K. We argue by contradiction. Consider a three-dimensional, c ompact and orientable solutiongij(t) to the Ricci ﬂow with normalized initial metric, a point ( x0,t0) and some radius r0>0 withr0< α, forα > 0 a suﬃciently small constant to be determined later, such that the assumptions of the theorem d o hold whereas the conclusion does not. We ﬁrst claim that we may assume that any other point ( x′,t′) and radius r′>0 with the same property has either t′>t0ort′<t0−2τr2 0, or 2r′> r0. Indeed, suppose otherwise. Then there exist ( x′ 0,t′ 0) andr′ 0witht′ 0∈[t0−2τr2 0,t0] andr′ 0≤1 2r0, for which the assumptions of the theorem hold but the conclu sion does not. Thus, there is a point ( x,t) such that t∈[t′ 0−τ(r′ 0)2,t′ 0]⊂/bracketleftig t0−2τr2 0−τ 4r2 0,t0/bracketrightig andR(x,t)>K(r′ 0)−2≥4Kr−2 0. If the point ( x′ 0,t′ 0) and the radius r′ 0satisfy the claim then we stop, and otherwise we iterate the procedure. Since t0≥4τr2 0and the solution is smooth, the iteration must terminate in a ﬁnite number of steps, which provides the desired point and the desired radius. Letτ′≥0 be the largest number such that (7.2.5) Rm(x,t)≥ −r−2 0 whenevert∈[t0−τ′r2 0,t0] anddt(x,x0)≤r0. Ifτ′≥2τ, we are done by Theorem 6.3.3 (ii). Thus we may assume τ′<2τ. By applying Theorem 6.3.3(ii), we know that at time t′=t0−τ′r2 0, the ballBt′(x0,r0) has (7.2.6) Vol t′(Bt′(x0,r0))≥ξ(w)r3 0 for some positive constant ξ(w) depending only on w. We next claim that there exists a ball (at time t′=t0−τ′r2 0)Bt′(x′,r′)⊂Bt′(x0,r0) with (7.2.7) Vol t′(Bt′(x′,r′))≥1 2α3(r′)3 and with (7.2.8)r0 2>r′≥c(w)r0 for some small positive constant c(w) depending only on w, whereα3is the volume of the unit ball B3in the Euclidean space R3. Indeed, suppose that it is not true. Then after rescaling, th ere is a sequence of Riemannian manifolds Mi, i= 1,2,...,with ballsB(xi,1)⊂Miso that (7.2.5)′Rm≥ −1 onB(xi,1) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 411 and (7.2.6)′Vol (B(xi,1))≥ξ(w) for alli, but all balls B(x′ i,r′ i)⊂B(xi,1) with1 2>r′ i≥1 isatisfy (7.2.9) Vol( B(x′ i,r′ i))<1 2α3(r′ i)3. It follows from basic results in Alexandrov space theory (se e for example Theorem 10.7.2 and Theorem 10.10.10 of [9]) that, after taking a subs equence, the marked balls (B(xi,1),xi) converge in the Gromov-Hausdorﬀ topology to a marked lengt h space (B∞,x∞) with curvature bounded from below by −1 in the Alexandrov space sense, and the associated Riemannian volume forms dVolMiover (B(xi,1),xi) con- verge weakly to the Hausdorﬀ measure µofB∞. It is well-known that the Hausdorﬀ dimension of any Alexandrov space is either an integer or inﬁ nity (see for example Theorem 10.8.2 of [9]). Then by (7.2.6)′, we know the limit ( B∞,x∞) is a three- dimensional Alexandrov space of curvature ≥ −1. In the Alexandrov space theory, a pointp∈B∞is said to be regular if the tangent cone of B∞atpis isometric to R3. It is also a basic result in Alexandrov space theory (see for e xample Corollary 10.9.13 of [9]) that the set of regular points in B∞is dense and for each regular point there is a small neighborhood which is almost isometric to an open s et of the Euclidean space R3. Thus for any ε>0, there are balls B(x′ ∞,r′ ∞)⊂B∞with 0<r′ ∞<1 3and satisfying µ(B(x′ ∞,r′ ∞))≥(1−ε)α3(r′ ∞)3. This is a contradiction with (7.2.9). Without loss of generality, we may assume w≤1 4α3. Sinceτ′<2τ, it follows from the choice of the point ( x0,t0) and the radius r0and (7.2.5), (7.2.7), (7.2.8) that the conclusion of the theorem holds for ( x′,t′) andr′. Thus we have the estimate R(x,t)≤K(r′)−2 whenevert∈[t′−τ(r′)2,t′] anddt(x,x′)≤1 4r′. Forα>0 small, by combining with the pinching estimate (7.1.1), we have |Rm(x,t)| ≤K′(r′)−2 whenevert∈[t′−τ(r′)2,t′] anddt(x,x′)≤1 4r′, whereK′is some positive constant depending only on K. Note that this curvature estimate implies the evolving met rics are equivalent over a suitable subregion of {(x,t)|t∈[t′−τ(r′)2,t′] anddt(x,x′)≤ 1 4r′}. Now we can apply Theorem 7.2.1 to choose α=α(w)>0 so small that (7.2.10) R(x,t)≤˜K(w)(r′)−2≤˜K(w)c(w)−2r−2 0 whenevert∈[t′−τ 2(r′)2,t′] anddt(x,x′)≤10r0. Then the combination of (7.2.10) with the pinching estimate (7.1.2) would imply Rm(x,t)≥ −[f−1(R(x,t)(1 +t))/(R(x,t)(1 +t))]R(x,t) ≥ −1 2r−2 0 412 H.-D. CAO AND X.-P. ZHU on the region {(x,t)|t∈[t′−τ 2(r′)2,t′] anddt(x,x0)≤r0}whenα=α(w)>r0small enough. This contradicts the choice of τ′. Therefore we have proved the theorem. The combination of the above two theorems immediately gives the following con- sequence. Corollary 7.2.3. For anyw >0andA <+∞, there exist τ=τ(w,A)>0, K=K(w,A)<+∞, andα=α(w,A)>0with the following property. Suppose we have a three-dimensional, compact and orientable solution to the Ricci ﬂow deﬁned onM×[0,T)with normalized initial metric. Suppose that for some radiu sr0>0 withr0<αand a point (x0,t0)∈M×[0,T)withT >t 0≥4τr2 0, the solution on the ballBt0(x0,r0)satisﬁes Rm(x,t0)≥ −r−2 0on B t0(x0,r0), and Volt0(Bt0(x0,r0))≥wr3 0. ThenR(x,t)≤Kr−2 0whenevert∈[t0−τr2 0,t0]anddt(x,x0)≤Ar0. We can also state the previous corollary in the following ver sion. Corollary 7.2.4 ( Perelman [103] ).For anyw >0one can ﬁnd ρ=ρ(w)>0 such that if gij(t)is a complete solution to the Ricci ﬂow deﬁned on M×[0,T)with T >1and with normalized initial metric, where Mis a three-dimensional, compact and orientable manifold, and if Bt0(x0,r0)is a metric ball at time t0≥1, withr0<ρ, such that min{Rm(x,t0)|x∈Bt0(x0,r0)}=−r−2 0, then Volt0(Bt0(x0,r0))≤wr3 0. Proof. We argue by contradiction. Suppose for any ρ>0, there is a solution and a ballBt0(x0,r0) satisfying the assumption of the corollary with r0<ρ,t0≥1, and with min{Rm(x,t0)|x∈Bt0(x0,r0)}=−r−2 0, but Volt0(Bt0(x0,r0))>wr3 0. We can apply Corollary 7.2.3 to get R(x,t)≤Kr−2 0 whenevert∈[t0−τr2 0,t0] anddt(x,x0)≤2r0, providedρ>0 is so small that 4 τρ2≤1 andρ<α , whereτ, αandKare the positive constants obtained in Corollary 7.2.3. Then forr0< ρandρ >0 suﬃciently small, it follows from the pinching estimate (7.1.2) that Rm(x,t)≥ −[f−1(R(x,t)(1 +t))/(R(x,t)(1 +t))]R(x,t) ≥ −1 2r−2 0 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 413 in the region {(x,t)|t∈[t0−τ(r0)2,t0] anddt(x,x0)≤2r0}. In particular, this would imply min{Rm(x,t0)|x∈Bt0(x0,r0)}>−r−2 0. This contradicts the assumption. 7.3. Ricci Flow with Surgery. One of the central themes of the Ricci ﬂow theory is to give a classiﬁcation of all compact orientable t hree-manifolds. As we mentioned before, the basic idea is to obtain long-time beha vior of solutions to the Ricci ﬂow. However the solutions will in general become sing ular in ﬁnite time. Fortunately, we now understand the precise structures of th e solutions around the singularities, thanks to Theorem 7.1.1. When a solution dev elops singularities, one can perform geometric surgeries by cutting oﬀ the canonical neighborhoods around the singularities and gluing back some known pieces, and the n continue running the Ricci ﬂow. By repeating this procedure, one hopes to get a kin d of “weak” solution. In this section we will give a detailed description of this su rgery procedure and deﬁne a global “weak” solution to the Ricci ﬂow. Given anyε>0, based on the singularity structure theorem (Theorem 7.1. 1), we can get a clear picture of the solution near the singular time as follows. Let (M,g ij(·,t)) be a maximal solution to the Ricci ﬂow on the maximal time interval [0,T) withT < +∞, whereMis a connected compact orientable three- manifold and the initial metric is normalized. For the given ε >0 and the solution (M,g ij(·,t)), we can ﬁnd r0>0 such that each point ( x,t) withR(x,t)≥r−2 0satisﬁes the derivative estimates (7.3.1) |∇R(x,t)|<ηR3 2(x,t) and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ ∂tR(x,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ηR2(x,t), whereη>0 is a universal constant, and has a canonical neighborhood w hich is either an evolving ε-neck, or an evolving ε-cap, or a compact positively curved manifold (without boundary). In the last case the solution becomes ex tinct at the maximal timeTand the manifold Mis diﬀeomorphic to the round three-sphere S3or a metric quotient of S3by Theorem 5.2.1. Let Ω denote the set of all points in Mwhere the curvature stays bounded as t→T. The gradient estimates in (7.3.1) imply that Ω is open and th atR(x,t)→ ∞ ast→Tfor eachx∈M\Ω. If Ω is empty, then the solution becomes extinct at time T. In this case, either the manifold Mis compact and positively curved, or it is entirely covered b y evolving ε-necks and evolving ε-caps shortly before the maximal time T. So the manifold M is diﬀeomorphic to either S3, or a metric quotient of the round S3, orS2×S1, or RP3#RP3. The reason is as follows. Clearly, we only need to consider t he situation that the manifold Mis entirely covered by evolving ε-necks and evolving ε-caps shortly before the maximal time T. IfMcontains a cap C, then there is a cap or a neck adjacent to the neck-like end of C. The former case implies that Mis diﬀeomorphic toS3,RP3, orRP3#RP3. In the latter case, we get a new longer cap and continue. Finally, we must end up with a cap, producing a S3,RP3, orRP3#RP3. IfMcontains no caps, we start with a neck N. By connecting with the necks that are adjacent to the boundary of N, we get a longer neck and continue. After a ﬁnite number of ste ps, the resulting neck must repeat itself. Since Mis orientable, we conclude that Mis diﬀeomorphic to S2×S1. 414 H.-D. CAO AND X.-P. ZHU We can now assume that Ω is nonempty. By using the local deriva tive estimates of Shi (Theorem 1.4.2), we see that as t→Tthe solution gij(·,t) has a smooth limit ¯gij(·) on Ω. Let ¯R(x) denote the scalar curvature of ¯ gij. For any ρ < r 0, let us consider the set Ωρ={x∈Ω|¯R(x)≤ρ−2}. By the evolution equation of the Ricci ﬂow, we see that the ini tial metricgij(·,0) and the limit metric gij(·) are equivalent over any ﬁxed region where the curvature rem ains uniformly bounded. Note that for any ﬁxed x∈∂Ω, and any sequence of points xj∈Ω withxj→xwith respect to the initial metric gij(·,0), we have R(xj)→+∞. In fact, if there were a subsequence xjkso that lim k→∞R(xjk) exists and is ﬁnite, then it would follow from the gradient estimates (7.3.1) that Ris uniformly bounded in some small neighborhood of x∈∂Ω (with respect to the induced topology of the initial metricgij(·,0)); this is a contradiction. From this observation and the c ompactness of the initial manifold, we see that Ω ρis compact (with respect to the metric gij(·)). For further discussions, let us introduce the following ter minologies. Denote by I an interval. Recall that an ε-neck (of radiusr) is an open set with a Riemannian metric, which is, after scaling the metric with factor r−2,ε-close to the standard neck S2×I with the product metric, where S2has constant scalar curvature one and Ihas length 2ε−1and theε-closeness refers to the C[ε−1]topology. A metric on S2×I, such that each point is contained in some ε-neck, is called an ε-tube , or anε-horn , or adoubleε-horn , if the scalar curvature stays bounded on both ends, or stays bounded on one end and tends to inﬁnity on t he other, or tends to inﬁnity on both ends, respectively. A metric on B3orRP3\¯B3is called aε-capif the region outside some suitable compact subset is an ε-neck. A metric on B3orRP3\¯B3is called an cappedε-horn if each point outside some compact subset is contained in an ε-neck and the scalar curvature tends to inﬁnity on the end. Now take any ε-neck in (Ω ,¯gij) and consider a point xon one of its boundary components. If x∈Ω\Ωρ, then there is either an ε-cap or anε-neck, adjacent to the initialε-neck. In the latter case we can take a point on the boundary of the second ε-neck and continue. This procedure can either terminate whe n we get into Ω ρor anε-cap, or go on indeﬁnitely, producing an ε-horn. The same procedure can be repeated for the other boundary component of the initial ε-neck. Therefore, taking into account that Ω has no compact components, we conclude th at eachε-neck of (Ω,¯gij) is contained in a subset of Ω of one of the following types: (a) anε-tube with boundary components in Ω ρ, or (b) anε-cap with boundary in Ω ρ, or (c) anε-horn with boundary in Ω ρ, or (7.3.2) (d) a capped ε-horn, or (e) a double ε-horn. Similarly, each ε-cap of (Ω,¯gij) is contained in a subset of Ω of either type (b) or type (d). It is clear that there is a deﬁnite lower bound (depending on ρ) for the volume of subsets of types (a), (b) and (c), so there can be only a ﬁnite n umber of them. Thus THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 415 we conclude that there is only a ﬁnite number of components of Ω containing points of Ω ρ, and every such component has a ﬁnite number of ends, each bei ng anε-horn. On the other hand, every component of Ω, containing no points of Ω ρ, is either a cappedε-horn, or a double ε-horn. If we look at the solution at a slightly earlier time, the above argument shows each ε-neck orε-cap of (M,g ij(·,t)) is contained in a subset of types (a) or (b), while the ε-horns, capped ε-horns and double ε-horns (at the maximal time T) are connected together to form ε-tubes and ε-caps at the times tshortly before T. Ωρ @RΩρ @R ε-horn@Iε-tube@I doubleε-horn6 cappedε-horn6 Hence, by looking at the solution at times shortly before T, we see that the topology of Mcan be reconstructed as follows: take the components Ω j, 1≤j≤k, of Ω which contain points of Ω ρ, truncate their ε-horns, and glue to the boundary components of truncated Ω ja ﬁnite collection of tubes S2×Iand caps B3orRP3\¯B3. Thus,Mis diﬀeomorphic to a connected sum of ¯Ωj, 1≤j≤k, with a ﬁnite number of copies of S2×S1(which correspond to gluing a tube to two boundary component s of the same Ω j), and a ﬁnite number of copies of RP3. Here ¯Ωjdenotes Ω jwith each ε-horn one point compactiﬁed. More geometrically, one can ge t¯Ωjin the following way: in every ε-horn of Ω jone can ﬁnd an ε-neck, cut it along the middle two-sphere, remove the horn-shaped end, and glue back a cap (or more preci sely, a diﬀerentiable three-ball). Thus to understand the topology of M, one only needs to understand the topologies of the compact orientable three-manifolds ¯Ωj, 1≤j≤k. Naturally one can evolve each ¯Ωjby the Ricci ﬂow again and, when singularities develop again, perform the above surgery for each ε-horn to get new compact ori- entable three-manifolds. By repeating this procedure inde ﬁnitely, it will likely give a long-time “weak” solution to the Ricci ﬂow. The following ab stract deﬁnition for this kind of “weak” solution was introduced by Perelman in [104] . Definition 7.3.1. Suppose we are given a (ﬁnite or countably inﬁnite) collecti on of three-dimensional smooth solutions gk ij(t) to the Ricci ﬂow deﬁned on Mk×[t− k,t+ k) 416 H.-D. CAO AND X.-P. ZHU and go singular as t→t+ k, where each manifold Mkis compact and orientable, possibly disconnected with only a ﬁnite number of connected componen ts. Let (Ω k,¯gk ij) be the limits of the corresponding solutions gk ij(t) ast→t+ k, as above. Suppose also that for eachkwe havet− k=t+ k−1, and that (Ω k−1,¯gk−1 ij) and (Mk,gk ij(t− k)) contain compact (possibly disconnected) three-dimensional submanifolds with smooth boundary which are isometric. Then by identifying these isometric submani folds, we say the collection of solutions gk ij(t) is a solution to the Ricci ﬂow with surgery (or asurgically modiﬁed solution to the Ricci ﬂow) on the time interval which is the union of all [t− k,t+ k), and say the times t+ karesurgery times . To get the topology of the initial manifold from the solution to the Ricci ﬂow with surgery, one has to overcome the following two diﬃculti es: (i) how to prevent the surgery times from accumulating? (ii) how to obtain the long time behavior of the solution to th e Ricci ﬂow with surgery? Thus it is natural to consider those solutions having “good” properties. For any arbitrarily ﬁxed positive number ε, we will only consider those solutions to the Ricci ﬂow with surgery which satisfy the following a priori assumptions (with accuracyε). Pinching assumption. The eigenvalues λ≥µ≥νof the curvature operator of the solution to the Ricci ﬂow with surgery at each point and each time satisfy (7.3.3) R≥(−ν)[log(−ν) + log(1 + t)−3] wheneverν <0. Canonical neighborhood assumption (with accuracy ε).For any given ε >0, there exist positive constants C1andC2depending only on ε, and a nonin- creasing positive function r: [0,+∞)→(0,+∞) such that at each time t>0, every pointxwhere scalar curvature R(x,t) is at least r−2(t) has a neighborhood B, with Bt(x,σ)⊂B⊂Bt(x,2σ) for some 0 <σ < C 1R−1 2(x,t), which falls into one of the following three categories: (a)Bis astrongε-neck (in the sense Bis the slice at time tof the parabolic neighborhood {(x′,t′)|x′∈B,t′∈[t−R(x,t)−1,t]}, where the solution is well deﬁned on the whole parabolic neighborhood and is, afte r scaling with factorR(x,t) and shifting the time to zero, ε-close (in the C[ε−1]topology) to the subset ( S2×I)×[−1,0] of the evolving standard round cylinder with scalar curvature 1 to S2and length 2 ε−1toIat time zero), or (b)Bis anε-cap, or (c)Bis a compact manifold (without boundary) of positive sectio nal curvature. Furthermore, the scalar curvature in Bat timetis betweenC−1 2R(x,t) andC2R(x,t), satisﬁes the gradient estimates (7.3.4) |∇R|<ηR3 2and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂R ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ηR2, and the volume of Bin case (a) and case (b) satisﬁes (C2R(x,t))−3 2≤Volt(B)≤εσ3. Hereηis a universal positive constant. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 417 Without loss of generality, we always assume the above const antsC1andC2are twice bigger than the corresponding constants C1(ε 2) andC2(ε 2) in Theorem 6.4.6 with the accuracyε 2. We remark that the above deﬁnition of the canonical neighbor hood assumption is slightly diﬀerent from that of Perelman in [104] in two asp ects: (1) it allows the parameterrto depend on time; (2) it also includes an volume upper bound f or the canonical neighborhoods of types (a) and (b). Arbitrarily given a compact orientable three-manifold wit h a Riemannian metric, by scaling, we may assume the Riemannian metric is normalize d. In the rest of this section and the next section, we will show the Ricci ﬂow w ith surgery, with the normalized metric as initial data, has a long-time solution which satisﬁes the above a priori assumptions and has only a ﬁnite number of surgery tim es at each ﬁnite time interval. The construction of the long-time solution will b e given by an induction argument. First, for the arbitrarily given compact orientable normal ized three-dimensional Riemannian manifold ( M,g ij(x)), the Ricci ﬂow with it as initial data has a maximal solutiongij(x,t) on a maximal time interval [0 ,T) withT > 1. It follows from Theorem 5.3.2 and Theorem 7.1.1 that the a priori assumption s (with accuracy ε) hold for the smooth solution on [0 ,T). IfT= +∞, we have the desired long time solution. Thus, without loss of generality, we may assume th e maximal time T <+∞ so that the solution goes singular at time T. Suppose that we have a solution to the Ricci ﬂow with surgery, with the nor- malized metric as initial data, satisfying the a priori assu mptions (with accuracy ε), deﬁned on [0 ,T) withT <+∞, going singular at time Tand having only a ﬁnite number of surgery times on [0 ,T). Let Ω denote the set of all points where the cur- vature stays bounded as t→T. As we have seen before, the canonical neighborhood assumption implies that Ω is open and that R(x,t)→ ∞ ast→Tfor allxlying outside Ω. Moreover, as t→T, the solution gij(x,t) has a smooth limit ¯ gij(x) on Ω. For someδ >0 to be chosen much smaller than ε, we letρ=δr(T) wherer(t) is the positive nonincreasing function in the deﬁnition of t he canonical neighborhood assumption. We consider the corresponding compact set Ωρ={x∈Ω|¯R(x)≤ρ−2}, where ¯R(x) is the scalar curvature of ¯ gij. If Ω ρis empty, the manifold (near the maximal time T) is entirely covered by ε-tubes,ε-caps and compact components with positive curvature. Clearly, the number of the compact comp onents is ﬁnite. Then in this case the manifold (near the maximal time T) is diﬀeomorphic to the union of a ﬁnite number of copies of S3, or metric quotients of the round S3, orS2×S1, or a connected sum of them. Thus when Ω ρis empty, the procedure stops here, and we say the solution becomes extinct . We now assume Ω ρis not empty. Then we know that every point x∈Ω\Ωρlies in one of the subsets of Ω listed in (7.3.2), or in a compact component with positive curvature, or in a compa ct component which is contained in Ω \Ωρand is diﬀeomorphic to either S3, orS2×S1orRP3#RP3. Note again that the number of the compact components is ﬁnite . Let us throw away all the compact components lying in Ω \Ωρand all the compact components with positive curvature, and then consider those components Ω j, 1≤j≤k, of Ω which contain points of Ω ρ. (We will consider those components of Ω \Ωρconsisting of cappedε-horns and double ε-horns later). We will perform surgical procedures, as we roughly described before, by ﬁnding an ε-neck in every horn of Ω j, 1≤j≤k, 418 H.-D. CAO AND X.-P. ZHU and then cutting it along the middle two-sphere, removing th e horn-shaped end, and gluing back a cap. In order to maintain the a priori assumptions with the same accuracy after the surgery, we will need to ﬁnd suﬃcient “ﬁne” necks in the ε-horns and to glue suﬃcient “ﬁne” caps. Note that δ >0 will be chosen much smaller than ε >0. The following lemma due to Perelman [104] gives us the “ﬁne” n ecks in the ε-horns. (At the ﬁrst sight, we should also cut oﬀ all those ε-tubes and ε-caps in the surgery procedure. However, in general we are not able to ﬁnd a “ﬁne” n eck in anε-tube or in anε-cap, and surgeries at “rough” ε-necks will certainly lose some accuracy. If we perform surgeries at the necks with some ﬁxed accuracy εin the high curvature region at each surgery time, then it is possible that the errors of su rgeries may accumulate to a certain amount so that at some later time we cannot recogn ize the structure of very high curvature regions. This prevents us from carrying out the whole process in ﬁnite time with a ﬁnite number of steps. This is the reason why we will only perform the surgeries at the ε-horns.) Lemma 7.3.2 ( Perelman [104] ).Given 0<ε≤1 100,0<δ<ε and0<T < +∞, there exists a radius 0<h<δρ , depending only on δandr(T), such that if we have a solution to the Ricci ﬂow with surgery, with a normalized me tric as initial data, satisfying the a priori assumptions (with accuracy ε),deﬁned on [0,T), going singular at timeTand having only a ﬁnite number of surgery times on [0,T), then for each pointxwithh(x) =¯R−1 2(x)≤hin anε-horn of (Ω,¯gij)with boundary in Ωρ, the neighborhood BT(x,δ−1h(x))/defines{y∈Ω|dT(y,x)≤δ−1h(x)}is a strong δ-neck (i.e., BT(x,δ−1h(x))×[T−h2(x),T]is, after scaling with factor h−2(x),δ-close (in the C[δ−1]topology )to the corresponding subset of the evolving standard round c ylinder S2×Rover the time interval [−1,0]with scalar curvature 1at time zero ). strongδ-neckΩρ Proof. We argue by contradiction. Suppose that there exists a seque nce of solu- tionsgk ij(·,t),k= 1,2,..., to the Ricci ﬂow with surgery, satisfying the a priori as- sumptions (with accuracy ε), deﬁned on [0 ,T) with limit metrics (Ωk,¯gk ij),k= 1,2,..., and pointsxk, lying inside an ε-horn of Ωkwith boundary in Ωk ρ, and having h(xk)→0 such that the neighborhoods BT(xk,δ−1h(xk)) ={y∈Ωk|dT(y,xk)≤δ−1h(xk)} are not strong δ-necks. Let/tildewidegk ij(·,t) be the solutions obtained by rescaling by the factor ¯R(xk) =h−2(xk) aroundxkand shifting the time Tto the new time zero. We now want to show that a subsequence of /tildewidegk ij(·,t),k= 1,2,..., converges to the evolving round cylinder, which will give a contradiction. Note that/tildewidegk ij(·,t),k= 1,2,...,are solutions modiﬁed by surgery. So, we cannot apply Hamilton’s compactness theorem directly since it is s tated only for smooth solutions. For each (unrescaled) surgical solution ¯ gk ij(·,t), we pick a point zk, with ¯R(zk) = 2C2 2(ε)ρ−2,in theε-horn of (Ωk,¯gk ij) with boundary in Ωk ρ, whereC2(ε) is the positive constant in the canonical neighborhood assump tion. From the deﬁnition THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 419 ofε-horn and the canonical neighborhood assumption, we know th at each point x lying inside the ε-horn of (Ωk,¯gk ij) withd¯gk ij(x,Ωk ρ)≥d¯gk ij(zk,Ωk ρ) has a strong ε-neck as its canonical neighborhood. Since h(xk)→0, eachxklies deeply inside an ε-horn. Thus for each positive A <+∞, the rescaled (surgical) solutions /tildewidegk ij(·,t) with the marked origins xkover the geodesic balls B/CTgk ij(·,0)(xk,A), centered at xkof radiiA (with respect to the metrics /tildewidegk ij(·,0)), will be smooth on some uniform (size) small time intervals for all suﬃciently large k, if the curvatures of the rescaled solutions /tildewidegk ij att= 0 inB/CTgk ij(·,0)(xk,A) are uniformly bounded. In such a situation, Hamilton’s compactness theorem is applicable. Then we can apply the sam e argument as in the second step of the proof of Theorem 7.1.1 to conclude that for eachA <+∞, there exists a positive constant C(A) such that the curvatures of the rescaled solutions /tildewidegk ij(·,t) at the new time 0 satisfy the estimate |/tildewideRmk|(y,0)≤C(A) wheneverd/CTgk ij(·,0)(y,xk)≤Aandk≥1; otherwise we would get a piece of a non-ﬂat nonnegatively curved metric cone as a blow-up limit, which c ontradicts Hamilton’s strong maximum principle. Moreover, by Hamilton’s compact ness theorem (Theorem 4.1.5), a subsequence of the rescaled solutions /tildewidegk ij(·,t) converges to a C∞ loclimit/tildewideg∞ ij(·,t), deﬁned on a spacetime set which is relatively open in the half spacetime {t≤0}and contains the time slice t= 0. By the pinching assumption, the limit is a complete manifold with nonnegative sectional curvature. Since xkwas contained in an ε-horn with boundary in Ωk ρand h(xk)/ρ→0, the limiting manifold has two ends. Thus, by Toponogov’s s plitting theorem, the limiting manifold admits a metric splitting Σ2×R, where Σ2is diﬀeo- morphic to the two-sphere S2becausexkwas the center of a strong ε-neck. By combining with the canonical neighborhood assumption (w ith accuracy ε), we see that the limit is deﬁned on the time interval [ −1,0] and isε-close to the evolving standard round cylinder. In particular, the scalar curvatu re of the limit at time t=−1 isε-close to 1/2. Sinceh(xk)/ρ→0, each point in the limiting manifold at time t=−1 also has a strongε-neck as its canonical neighborhood. Thus the limit is deﬁne d at least on the time interval [ −2,0] and the limiting manifold at time t=−2 is, after rescaling, ε-close to the standard round cylinder. By using the canonical neighborhood assumption again, ever y point in the limiting manifold at time t=−2 still has a strong ε-neck as its canonical neighborhood. Also note that the scalar curvature of the limit at t=−2 is not bigger than 1 /2+ε. Thus the limit is deﬁned at least on the time interval [ −3,0] and the limiting manifold at timet=−3 is, after rescaling, ε-close to the standard round cylinder. By repeating this argument we prove that the limit exists on the ancient ti me interval ( −∞,0]. The above argument also shows that at every time, each point o f the limit has a strongε-neck as its canonical neighborhood. This implies that the l imit isκ- noncollaped on all scales for some κ >0. Therefore, by Theorem 6.2.2, the limit is the evolving round cylinder S2×R, which gives the desired contradiction. In the above lemma, the property that the radius hdepends only on δand the timeTbut is independent of the surgical solution is crucial; othe rwise we will not be able to cut oﬀ enough volume at each surgery to guarantee the n umber of surgeries being ﬁnite in each ﬁnite time interval. We also remark that t he above proof actually 420 H.-D. CAO AND X.-P. ZHU proves a stronger result: the parabolic region {(y,t)|y∈BT(x,δ−1h(x)),t∈[T− δ−2h2(x),T]}is, after scaling with factor h−2(x),δ-close (in the C[δ−1]topology) to the corresponding subset of the evolving standard round cyl inder S2×Rover the time interval [ −δ−2,0] with scalar curvature 1 at the time zero. This fact will be u sed later in the proof of Proposition 7.4.1. We next want to construct “ﬁne” caps. Take a rotationally sym metric metric onR3with nonnegative sectional curvature and positive scalar c urvature such that outside some compact set it is a semi-inﬁnite standard round cylinder (i.e. the metric product of a ray with the round two-sphere of scalar curvatur e 1). We call such a metric on R3astandard capped inﬁnite cylinder . By the short-time existence theorem of Shi (Theorem 1.2.3), the Ricci ﬂow with a standard capped inﬁnite cylinder as initial data has a complete solution on a maximal time inte rval [0,T) such that the curvature of the solution is bounded on R3×[0,T′] for each 0 <T′<T. Such a solution is called a standard solution by Perelman [104]. The following result proved by Chen and the second author in [ 34] gives the curvature estimate for standard solutions. Proposition 7.3.3. Letgijbe a complete Riemannian metric on Rn(n≥3) with nonnegative curvature operator and positive scalar cu rvature which is asymptotic to a round cylinder of scalar curvature 1at inﬁnity. Then there is a complete solution gij(·,t)to the Ricci ﬂow, with gijas initial metric, which exists on the time interval [0,n−1 2), has bounded curvature at each time t∈[0,n−1 2), and satisﬁes the estimate R(x,t)≥C−1 n−1 2−t for someCdepending only on the initial metric gij. Proof. Since the initial metric has bounded curvature operator and a positive lower bound on its scalar curvature, the Ricci ﬂow has a solut iongij(·,t) deﬁned on a maximal time interval [0 ,T) withT <∞which has bounded curvature on Rn×[0,T′] for each 0 < T′< T. By Proposition 2.1.4, the solution gij(·,t) has nonnegative curvature operator for all t∈[0,T). Note that the injectivity radius of the initial metric has a positive lower bound. As we remarked at the beginning of Section 3.4, the same proof of Perelman’s no local collapsin g theorem I concludes that gij(·,t) isκ-noncollapsed on all scales less than√ Tfor someκ >0 depending only on the initial metric. We will ﬁrst prove the following assertion. Claim 1. There is a positive function ω: [0,∞)→[0,∞) depending only on the initial metric and κsuch that R(x,t)≤R(y,t)ω(R(y,t)d2 t(x,y)) for allx,y∈Rn,t∈[0,T). The proof is similar to that of Theorem 6.4.3. Notice that the initial metric has nonnegative curvature operator and its scalar curvature sa tisﬁes the bounds (7.3.5) C−1/lessorequalslantR(x)/lessorequalslantC for some positive constant C. By the maximum principle, we know T≥1 2nCand R(x,t)≤2Cfort∈[0,1 4nC]. The assertion is clearly true for t∈[0,1 4nC]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 421 Now ﬁx (y,t0)∈Rn×[0,T) witht0≥1 4nC. Letzbe the closest point to ywith the property R(z,t0)d2 t0(z,y) = 1 (at time t0). Draw a shortest geodesic from ytoz and choose a point ˜ zon the geodesic satisfying dt0(z,˜z) =1 4R(z,t0)−1 2, then we have R(x,t0)≤1 (1 2R(z,t0)−1 2)2onBt0/parenleftbigg ˜z,1 4R(z,t0/parenrightbigg−1 2 ). Note that R(x,t)/greaterorequalslantC−1everywhere by the evolution equation of the scalar curvature. Then by the Li-Yau-Hamilton inequality (Coroll ary 2.5.5), for all ( x,t)∈ Bt0(˜z,1 8nCR(z,t0)−1 2)×[t0−(1 8nCR(z,t0)−1 2)2,t0], we have R(x,t)≤ t0 t0−/parenleftig 1 8n√ C/parenrightig2 1 /parenleftig 1 2R(z,t0)−1 2/parenrightig2 ≤/bracketleftbigg1 8nCR(z,t0)−1 2/bracketrightbigg−2 . Combining this with the κ-noncollapsing property, we have Vol/parenleftbigg Bt0/parenleftbigg ˜z,1 8nCR(z,t0)−1 2/parenrightbigg/parenrightbigg ≥κ/parenleftbigg1 8nCR(z,t0)−1 2/parenrightbiggn and then Vol/parenleftig Bt0/parenleftig z,8R(z,t0)−1 2/parenrightig/parenrightig ≥κ/parenleftbigg1 64nC/parenrightbiggn/parenleftig 8R(z,t0)−1 2/parenrightign . So by Theorem 6.3.3 (ii), we have R(x,t0)≤C(κ)R(z,t0) for allx∈Bt0/parenleftig z,4R(z,t0)−1 2/parenrightig . Here and in the following we denote by C(κ) various positive constants depending only onκ,nand the initial metric. Now by the Li-Yau-Hamilton inequality (Corollary 2.5.5) an d local gradient esti- mate of Shi (Theorem 1.4.2), we obtain R(x,t)≤C(κ)R(z,t0) and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ ∂tR/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x,t)≤C(κ)(R(z,t0))2 for all (x,t)∈Bt0(z,2R(z,t0)−1 2))×[t0−(1 8nCR(z,t0)−1 2)2,t0]. Therefore by com- bining with the Harnack estimate (Corollary 2.5.7), we obta in R(y,t0)≥C(κ)−1R(z,t0−C(κ)−1R(z,t0)−1) ≥C(κ)−2R(z,t0) Consequently, we have showed that there is a constant C(κ) such that Vol/parenleftig Bt0/parenleftig y,R(y,t0)−1 2/parenrightig/parenrightig ≥C(κ)−1/parenleftig R(y,t0)−1 2/parenrightign and R(x,t0)≤C(κ)R(y,t0) for allx∈Bt0/parenleftig y,R(y,t0)−1 2/parenrightig . 422 H.-D. CAO AND X.-P. ZHU In general, for any r≥R(y,t0)−1 2, we have Vol(Bt0(y,r))≥C(κ)−1(r2R(y,t0))−n 2rn. By applying Theorem 6.3.3(ii) again, there exists a positiv e constant ω(r2R(y,t0)) depending only on the constant r2R(y,t0) andκsuch that R(x,t0)≤R(y,t0)ω(r2R(y,t0)) for all x∈Bt0/parenleftbigg y,1 4r/parenrightbigg . This proves the desired Claim 1. Now we study the asymptotic behavior of the solution at inﬁni ty. For any 0 < t0< T, we know that the metrics gij(x,t) witht∈[0,t0] has uniformly bounded curvature. Let xkbe a sequence of points with d0(x0,xk)→ ∞. After passing to a subsequence, gij(x,t) aroundxkwill converge to a solution to the Ricci ﬂow on R×Sn−1with round cylinder metric of scalar curvature 1 as initial d ata. Denote the limit by ˜gij. Then by the uniqueness theorem (Theorem 1.2.4), we have ˜R(x,t) =n−1 2 n−1 2−tfor allt∈[0,t0]. It follows that T≤n−1 2. In order to show T=n−1 2, it suﬃces to prove the following assertion. Claim 2. SupposeT <n−1 2. Fix a point x0∈Rn, then there is a δ >0, such that for any x∈Mwithd0(x,x0)≥δ−1, we have R(x,t)≤2C+n−1 n−1 2−tfor allt∈[0,T), whereCis the constant in (7.3.5). In view of Claim 1, if Claim 2 holds, then sup Mn×[0,T)R(y,t)≤ω/parenleftbigg δ−2/parenleftbigg 2C+n−1 n−1 2−T/parenrightbigg/parenrightbigg/parenleftbigg 2C+n−1 n−1 2−T/parenrightbigg <∞ which will contradict the deﬁnition of T. To show Claim 2, we argue by contradiction. Suppose for each δ >0, there is a point (xδ,tδ) with 0<tδ<Tsuch that R(xδ,tδ)>2C+n−1 n−1 2−tδandd0(xδ,x0)≥δ−1. Let ¯tδ= sup/braceleftigg t/vextendsingle/vextendsingle/vextendsingle sup Mn\B0(x0,δ−1)R(y,t)<2C+n−1 n−1 2−t/bracerightigg . Since lim d0(y,x0)→∞R(y,t) =n−1 2 n−1 2−tand supM×[0,1 4nC]R(y,t)≤2C, we know1 4nC≤¯tδ≤ tδand there is a ¯ xδsuch thatd0(x0,¯xδ)≥δ−1andR(¯xδ,¯tδ) = 2C+n−1 n−1 2−¯tδ.By Claim THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 423 1 and Hamilton’s compactness theorem (Theorem 4.1.5), for δ→0 and after taking a subsequence, the metrics gij(x,t) onB0(¯xδ,δ−1 2) over the time interval [0 ,¯tδ] will converge to a solution ˜ gijonR×Sn−1with the standard metric of scalar curvature 1 as initial data over the time interval [0 ,¯t∞], and its scalar curvature satisﬁes ˜R(¯x∞,¯t∞) = 2C+n−1 n−1 2−¯t∞, ˜R(x,t)/lessorequalslant2C+n−1 n−1 2−¯t∞,for allt∈[0,¯t∞], where (¯x∞,¯t∞) is the limit of (¯ xδ,¯tδ). On the other hand, by the uniqueness theorem (Theorem 1.2.4) again, we know ˜R(¯x∞,¯t∞) =n−1 2 n−1 2−¯t∞ which is a contradiction. Hence we have proved Claim 2 and the n have veriﬁed T=n−1 2. Now we are ready to show (7.3.6) R(x,t)≥˜C−1 n−1 2−t,for all (x,t)∈Rn×/bracketleftig 0,n−1 2/parenrightig , for some positive constant ˜Cdepending only on the initial metric. For any (x,t)∈Rn×[0,n−1 2), by Claim 1 and κ-noncollapsing, there is a constant C(κ)>0 such that Volt(Bt(x,R(x,t)−1 2))≥C(κ)−1(R(x,t)−1 2)n. Then by the well-known volume estimate of Calabi-Yau (see fo r example [128] or [112]) for complete manifolds with Ric ≥0, for anya≥1, we have Volt(Bt(x,aR(x,t)−1 2))≥C(κ)−1a 8n(R(x,t)−1 2)n. On the other hand, since ( Rn,gij(·,t)) is asymptotic to a cylinder of scalar curvature (n−1 2)/(n−1 2−t), for suﬃciently large a>0, we have Volt/parenleftigg Bt/parenleftigg x,a/radicalbigg n−1 2−t/parenrightigg/parenrightigg ≤C(n)a/parenleftbiggn−1 2−t/parenrightbiggn 2 . Combining the two inequalities, for all suﬃciently large a, we have: C(n)a/parenleftbiggn−1 2−t/parenrightbiggn 2 ≥Volt Bt x,a /radicalig n−1 2−t R(x,t)−1 2 R(x,t)−1 2   ≥C(κ)−1a 8n /radicalig n−1 2−t R(x,t)−1 2 /parenleftig R(x,t)−1 2/parenrightign , which gives the desired estimate (7.3.6). Therefore the pro of of the proposition is complete. 424 H.-D. CAO AND X.-P. ZHU We now ﬁx a standard capped inﬁnite cylinder for dimension n= 3 as follows. Consider the semi-inﬁnite standard round cylinder N0=S2×(−∞,4) with the metric g0of scalar curvature 1. Denote by zthe coordinate of the second factor ( −∞,4). Letfbe a smooth nondecreasing convex function on ( −∞,4) deﬁned by (7.3.7)  f(z) = 0, z≤0, f(z) =ce−P z, z∈(0,3], f(z) is strictly convex on z∈[3,3.9], f(z) =−1 2log(16 −z2), z∈[3.9,4), where the (small) constant c>0 and (big) constant P >0 will be determined later. Let us replace the standard metric g0on the portion S2×[0,4) of the semi-inﬁnite cylinder by ˆ g=e−2fg0. Then the resulting metric ˆ gwill be smooth on R3obtained by adding a point to S2×(−∞,4) atz= 4. We denote by C(c,P) = (R3,ˆg). Clearly, C(c,P) is a standard capped inﬁnite cylinder. We next use a compact portion of the standard capped inﬁnite c ylinderC(c,P) and theδ-neck obtained in Lemma 7.3.2 to perform the following surge ry procedure due to Hamilton [64]. Consider the metric ¯ gat the maximal time T < +∞. Take an ε-horn with boundary in Ω ρ. By Lemma 7.3.2, there exists a δ-neckNof radius 0<h<δρ in the ε-horn. By deﬁnition, ( N,h−2¯g) isδ-close (in the C[δ−1]topology) to the standard round neck S2×Iof scalar curvature 1 with I= (−δ−1,δ−1). Using the parameter z∈I, we see the above function fis deﬁned on the δ-neckN. Let us cut the δ-neckNalong the middle (topological) two-sphere N/intersectiontext{z= 0}. Without loss of generality, we may assume that the right han d half portion N/intersectiontext{z≥0}is contained in the horn-shaped end. Let ϕbe a smooth bump function withϕ= 1 forz≤2, andϕ= 0 forz≥3. Construct a new metric ˜ gon a (topological) three-ball B3as follows (7.3.8) ˜ g=  ¯g, z = 0, e−2f¯g, z ∈[0,2], ϕe−2f¯g+ (1−ϕ)e−2fh2g0, z ∈[2,3], h2e−2fg0, z ∈[3,4]. The surgery is to replace the horn-shaped end by the cap ( B3,˜g). We call such surgery procedure a δ-cutoﬀ surgery . The following lemma determines the constants candPin theδ-cutoﬀ surgery so that the pinching assumption is preserved under the surgery . Lemma 7.3.4 ( Justiﬁcation of the pinching assumption ).There are universal positive constants δ0,c0andP0such that if we take a δ-cutoﬀ surgery at a δ-neck of radiushat timeTwithδ≤δ0andh−2≥2e2log(1 +T), then we can choose c=c0 andP=P0in the deﬁnition of f(z)such that after the surgery, there still holds the pinching condition (7.3.9) ˜R≥(−˜ν)[log(−˜ν) + log(1 + T)−3] whenever ˜ν <0, where ˜Ris the scalar curvature of the metric ˜gand˜νis the least eigenvalue of the curvature operator of ˜g. Moreover, after the surgery, any metric ball THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 425 of radiusδ−1 2hwith center near the tip (i.e., the origin of the attached cap )is, after scaling with factor h−2,δ1 2-close (in theC[δ−1 2]topology )to the corresponding ball of the standard capped inﬁnite cylinder C(c0,P0). Proof. First, we consider the metric ˜ gon the portion {0≤z≤2}. Under the conformal change ˜ g=e−2f¯g, the curvature tensor ˜Rijklis given by ˜Rijkl=e−2f/bracketleftig ¯Rijkl+|¯∇f|2(¯gil¯gjk−¯gik¯gjl) + (fik+fifk)¯gjl + (fjl+fjfl)¯gik−(fil+fifl)¯gjk−(fjk+fjfk)¯gil/bracketrightig . If{¯Fa=¯Fi a∂ ∂xi}is an orthonormal frame for ¯ gij, then {˜Fa=ef¯Fa=˜Fi a∂ ∂xi}is an orthonormal frame for ˜ gij. Let ¯Rabcd=¯Rijkl¯Fi a¯Fj b¯Fk c¯Fl d, ˜Rabcd=˜Rijkl˜Fi a˜Fj b˜Fk c˜Fl d, then ˜Rabcd=e2f/bracketleftig ¯Rabcd+|¯∇f|2(δadδbc−δacδbd) + (fac+fafc)δbd (7.3.10) + (fbd+fbfd)δac−(fad+fafd)δbc−(fbc+fbfc)δad/bracketrightig , and (7.3.11) ˜R=e2f(¯R+ 4¯△f−2|¯∇f|2). Since df dz=ce−P zP z2,d2f dz2=ce−P z/parenleftbiggP2 z4−2P z3/parenrightbigg , then for any small θ >0, we may choose c >0 small and P >0 large such that for z∈[0,3], we have (7.3.12) |e2f−1|+/vextendsingle/vextendsingle/vextendsingle/vextendsingledf dz/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggdf dz/parenrightbigg2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<θd2f dz2,d2f dz2<θ. On the other hand, by the deﬁnition of δ-neck of radius h, we have |¯g−h2g0|g0<δh2, |o ∇j¯g|g0<δh2,for 1≤j≤[δ−1], whereg0is the standard metric of the round cylinder S2×R. Note that in three dimensions, we can choose the orthonormal frame {¯F1,¯F2,¯F3}for the metric ¯ gso that its curvature operator is diagonal in the orthonormal f rame{√ 2¯F2∧¯F3,√ 2¯F3∧ ¯F1,√ 2¯F1∧¯F2}with eigenvalues ¯ ν≤¯µ≤¯λand ¯ν= 2¯R2323,¯µ= 2¯R3131,¯λ= 2¯R1212. 426 H.-D. CAO AND X.-P. ZHU Sinceh−2¯gisδ-close to the standard round cylinder metric g0on theδ-neck, we have (7.3.13)  |¯R3131|+|¯R2323|<δ7 8h−2, |¯R1212−1 2h−2|<δ7 8h−2, |¯F3−h−1∂ ∂z|g0<δ7 8h−1, for suitably small δ >0. Since ¯∇az=¯∇z(¯Fa) and ¯∇a¯∇bz=¯∇2z(¯Fa,¯Fb), it follows that |¯∇3z−h−1|<δ7 8h−1, |¯∇1z|+|¯∇2z|<δ7 8h−1, and |¯∇a¯∇bz|<δ7 8h−2,for 1≤a,b≤3. By combining with ¯∇af=df dz¯∇az,¯∇a¯∇bf=df dz¯∇a¯∇bz+d2f dz2¯∇az¯∇bz and (7.3.12), we get (7.3.14)  |¯∇af|<2θh−1d2f dz2, for 1≤a≤3, |¯∇a¯∇bf|<δ3 4h−2d2f dz2, unlessa=b= 3, |¯∇3¯∇3f−h−2d2f dz2|<δ3 4h−2d2f dz2. By combining (7.3.10) and (7.3.14), we have (7.3.15)  ˜R1212≥¯R1212−(θ1 2+δ5 8)h−2d2f dz2, ˜R3131≥¯R3131+ (1−θ1 2−δ5 8)h−2d2f dz2, ˜R2323≥¯R2323+ (1−θ1 2−δ5 8)h−2d2f dz2, |˜Rabcd| ≤(θ1 2+δ5 8)h−2d2f dz2,otherwise, whereθandδare suitably small. Then it follows that ˜R≥¯R+ [4−6(θ1 3+δ1 2)]h−2d2f dz2, −˜ν≤ −¯ν−[2−2(θ1 3+δ1 2)]h−2d2f dz2, for suitably small θandδ. If 0<−˜ν≤e2, then by the assumption that h−2≥2e2log(1 +T), we have ˜R≥¯R ≥1 2h−2 ≥e2log(1 +T) ≥(−˜ν)[log(−˜ν) + log(1 + T)−3]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 427 While if −˜ν >e2, then by the pinching estimate of ¯ g, we have ˜R≥¯R ≥(−¯ν)[log(−¯ν) + log(1 + T)−3] ≥(−˜ν)[log(−˜ν) + log(1 + T)−3]. So we have veriﬁed the pinching condition on the portion {0≤z≤2}. Next, we consider the metric ˜ gon the portion {2≤z≤4}. Letθbe a ﬁxed suitably small positive number. Then the constant c=c0andP=P0are ﬁxed. Soζ= min z∈[1,4]d2f dz2>0 is also ﬁxed. By the same argument as in the derivation of (7.3.15) from (7.3.10), we see that the curvature of the me tric ˆg=e−2fg0of the standard capped inﬁnite cylinder C(c0,P0) on the portion {1≤z≤4}is bounded from below by2 3ζ >0. Sinceh−2¯gisδ-close to the standard round metric g0, the metrich−2˜gdeﬁned by (7.3.8) is clearly δ3 4-close to the metric ˆ g=e−2fg0of the standard capped inﬁnite cylinder on the portion {1≤z≤4}. Thus asδis suﬃciently small, the curvature operator of ˜ gon the portion {2≤z≤4}is positive. Hence the pinching condition (7.3.9) holds trivially on the portion {2≤z≤4}. The last statement in Lemma 7.3.4 is obvious from the deﬁniti on (7.3.8). Recall from Lemma 7.3.2 that the δ-necks at a time t>0, where we performed Hamilton’s surgeries, have their radii 0 <h<δρ =δ2r(t). Without loss of generality, we may assume the positive nonincreasing function r(t) in the deﬁnition of the canon- ical neighborhood assumption is less than 1 and the universa l constantδ0in Lemma 7.3.4 is also less than 1. We deﬁne a positive function ¯δ(t) by (7.3.16) ¯δ(t) = min/braceleftbigg1 2e2log(1 +t),δ0/bracerightbigg fort∈[0,+∞). From now on, we always assume 0 < δ < ¯δ(t) for anyδ-cutoﬀ surgery at time t >0 and assume c=c0andP=P0. As a result, the standard capped inﬁnite cylinder and the standard solution are also ﬁxed. The follow ing lemma, which will be used in the next section, gives the canonical neighborhoo d structure for the ﬁxed standard solution. Lemma 7.3.5. Letgij(x,t)be the above ﬁxed standard solution to the Ricci ﬂow onR3×[0,1). Then for any ε >0, there is a positive constant C(ε)such that each point (x,t)∈R3×[0,1)has an open neighborhood B, withBt(x,r)⊂B⊂Bt(x,2r) for some 0<r<C (ε)R(x,t)−1 2, which falls into one of the following two categories: either (a)Bis anε-cap, or (b)Bis anε-neck and it is the slice at the time tof the parabolic neighborhood Bt(x,ε−1R(x,t)−1 2)×[t−min{R(x,t)−1,t},t], on which the standard solu- tion is, after scaling with the factor R(x,t)and shifting the time tto zero, ε-close (in theC[ε−1]topology )to the corresponding subset of the evolving standard cylinder S2×Rover the time interval [−min{tR(x,t),1},0]with scalar curvature 1at the time zero. Proof. The proof of the lemma is reduced to two assertions. We now sta te and prove the ﬁrst assertion which takes care of those points wit h times close to 1. 428 H.-D. CAO AND X.-P. ZHU Assertion 1. For anyε>0, there is a positive number θ=θ(ε) with 0<θ< 1 such that for any ( x0,t0)∈R3×[θ,1), the standard solution on the parabolic neigh- borhoodBt0(x,ε−1R(x0,t0)−1 2)×[t0−ε−2R(x0,t0)−1,t0] is well-deﬁned and is, after scaling with the factor R(x0,t0),ε-close (in the C[ε−1]topology) to the corresponding subset of some orientable ancient κ-solution. We argue by contradiction. Suppose Assertion 1 is not true, t hen there exist ¯ε >0 and a sequence of points ( xk,tk) withtk→1, such that for each k, the standard solution on the parabolic neighborhood Btk(xk,¯ε−1R(xk,tk)−1 2)×[tk−¯ε−2R(xk,tk)−1,tk] is not, after scaling by the factor R(xk,tk), ¯ε-close to the corresponding subset of any ancient κ-solution. Note that by Proposition 7.3.3, there is a consta ntC >0 (depending only on the initial metric, hence it is universal ) such that R(x,t)≥ C−1/(1−t). This implies ¯ε−2R(xk,tk)−1≤C¯ε−2(1−tk)<tk, and then the standard solution on the parabolic neighborhoo dBtk(xk, ¯ε−1R(xk,tk)−1 2)×[tk−¯ε−2R(xk,tk)−1,tk] is well-deﬁned for klarge. By Claim 1 in the proof of Proposition 7.3.3, there is a positive functi onω: [0,∞)→[0,∞) such that R(x,tk)≤R(xk,tk)ω(R(xk,tk)d2 tk(x,xk)) for allx∈R3. Now by scaling the standard solution gij(·,t) aroundxkwith the factorR(xk,tk) and shifting the time tkto zero, we get a sequence of the rescaled solutions ˜gk ij(x,˜t) =R(xk,tk)gij(x,tk+˜t/R(xk,tk)) to the Ricci ﬂow deﬁned on R3 with˜t∈[−R(xk,tk)tk,0]. We denote the scalar curvature and the distance of the rescaled metric ˜ gk ijby˜Rkand˜d. By combining with Claim 1 in the proof of Proposition 7.3.3 and the Li-Yau-Hamilton inequality, we get ˜Rk(x,0)≤ω(˜d2 0(x,xk)) ˜Rk(x,˜t)≤R(xk,tk)tk ˜t+R(xk,tk)tkω(˜d2 0(x,xk)) for anyx∈R3and˜t∈(−R(xk,tk)tk,0]. Note that R(xk,tk)tk→ ∞ by Proposition 7.3.3. We have shown in the proof of Proposition 7.3.3 that th e standard solution is κ- noncollapsed on all scales less than 1 for some κ>0. Then from the κ-noncollapsing property, the above curvature estimates and Hamilton’s com pactness theorem, we know ˜gk ij(x,˜t) has a convergent subsequence (as k→ ∞) whose limit is an ancient, κ- noncollapsed, complete and orientable solution with nonne gative curvature operator. This limit must have bounded curvature by the same proof of St ep 3 in the proof of Theorem 7.1.1. This gives a contradiction. Hence Assertion 1 is proved. We now ﬁx the constant θ(ε) obtained in Assertion 1. Let Obe the tip of the standard capped inﬁnite cylinder R3(it is rotationally symmetric about Oat time 0, and it remains so as t>0 by the uniqueness Theorem 1.2.4). Assertion 2. There are constants B1(ε) andB2(ε) depending only on ε, such that if (x0,t0)∈R3×[0,θ) withdt0(x0,O)≤B1(ε), then there is a 0 < r < B 2(ε) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 429 such thatBt0(x0,r) is anε-cap; if (x0,t0)∈R3×[0,θ) withdt0(x0,O)≥B1(ε), then the parabolic neighborhood Bt0(x0,ε−1R(x0,t0)−1 2)×[t0−min{R(x0,t0)−1,t0},t0] is after scaling with the factor R(x0,t0) and shifting the time t0to zero,ε-close (in the C[ε−1]topology) to the corresponding subset of the evolving stand ard cylinder S2×R over the time interval [ −min{t0R(x0,t0),1},0] with scalar curvature 1 at time zero. Since the standard solution exists on the time interval [0 ,1), there is a con- stantB0(ε) such that the curvatures on [0 ,θ(ε)] are uniformly bounded by B0(ε). This implies that the metrics in [0 ,θ(ε)] are equivalent. Note that the initial met- ric is asymptotic to the standard capped inﬁnite cylinder. F or any sequence of pointsxkwithd0(O,x k)→ ∞ , after passing to a subsequence, gij(x,t) around xkwill converge to a solution to the Ricci ﬂow on R×S2with round cylinder metric of scalar curvature 1 as initial data. By the uniquene ss theorem (Theorem 1.2.4), the limit solution must be the standard evolving rou nd cylinder. This implies that there is a constant B1(ε)>0 depending on εsuch that for any ( x0,t0) with t0≤θ(ε) anddt0(x0,O)≥B1(ε), the standard solution on the parabolic neighbor- hoodBt0(x0,ε−1R(x0,t0)−1 2)×[t0−min{R(x0,t0)−1,t0},t0] is, after scaling with the factorR(x0,t0),ε-close to the corresponding subset of the evolving round cyl inder. Since the solution is rotationally symmetric around O, the cap neighborhood struc- tures of those points x0withdt0(x0,O)≤B1(ε) follow directly. Hence Assertion 2 is proved. The combination of these two assertions proves the lemma. Since there are only a ﬁnite number of horns with the other end connected to Ω ρ, we perform only a ﬁnite number of such δ-cutoﬀ surgeries at time T. Besides those horns, there could be capped horns and double horns which lie in Ω\Ωρ. As explained before, they are connected to form tubes or capped tubes at an y time slightly before T. Sowe can regard the capped horns and double horns (of Ω\Ωρ) to be extinct and throw them away at time T. We only need to remember that the connected sums were broken there. Remember that we have thrown away all compact components, either lying in Ω\Ωρor with positive sectional curvature , each of which is diﬀeomorphic to either S3, or a metric quotient of S3, orS2×S1orRP3#RP3. So we have also removed a ﬁnite number of copies of S3, or metric quotients of S3, orS2×S1orRP3#RP3at the timeT. Let us agree to declare extinct every compact component either with positive sectional curvature or lying in Ω\Ωρ; in particular, this allows us to exclude the components with positive sectional curvatur e from the list of canonical neighborhoods. In summary, our surgery at time Tconsists of the following four procedures: (1)performδ-cutoﬀ surgeries for all ε-horns, whose other ends are connected to Ωρ; (2)declare extinct every compact component which has positive sectional curva- ture; (3)throw away all capped horns and double horns lying in Ω\Ωρ; (4)declare extinct all compact components lying in Ω\Ωρ. (In Sections 7.6and7.7,we will add one more procedure by declaring extinct every compact component which has nonnegative scalar curvature. ) By Lemma 7.3.4, after performing surgeries at time T, the pinching assumption (7.3.3) still holds for the surgically modiﬁed manifold. Wi th this surgically modiﬁed manifold (possibly disconnected) as initial data, we now co ntinue our solution under 430 H.-D. CAO AND X.-P. ZHU the Ricci ﬂow until it becomes singular again at some time T′(> T). Therefore, we have extended the solution to the Ricci ﬂow with surgery, ori ginally deﬁned on [0 ,T) withT <+∞, to the new time interval [0 ,T′) withT′>T. By the proof of Theorem 5.3.2, we see that the solution to the Ricci ﬂow with surgery a lso satisﬁes the pinching assumption on [0 ,T′). It remains to verify the canonical neighborhood assumpti on (with accuracy ε) for the solution on the time interval [ T,T′) and to prove that this extension procedure works indeﬁnitely (unless it becomes e xtinct at some ﬁnite time) and that there exists at most a ﬁnite number of surgeries at ev ery ﬁnite time interval. We leave these arguments to the next section. Before we end this section, we check the following two assert ions of Perelman in [104] which will be used in the next section to estimate the Li -Yau-Perelman distance of space-time curves which stretch to surgery regions. Lemma 7.3.6 ( Perelman [104] ).For any 0< ε≤1/100,1< A < +∞and 0<θ< 1, one can ﬁnd ¯δ=¯δ(A,θ,ε )with the following property. Suppose we have a solution to the Ricci ﬂow with surgery which satisﬁes the a pr iori assumptions (with accuracyε)on[0,T]and is obtained from a compact orientable three-manifold by a ﬁnite number of δ-cutoﬀ surgeries with each δ<¯δ. Suppose we have a cutoﬀ surgery at timeT0∈(0,T), letx0be any ﬁxed point on the gluing caps (i.e., the regions aﬀected by the cutoﬀ surgeries at time T0),and letT1=min{T,T0+θh2}, whereh is the cutoﬀ radius around x0obtained in Lemma 7.3.2.Then either (i) the solution is deﬁned on P(x0,T0,Ah,T 1−T0)/defines{(x,t)|x∈Bt(x0,Ah),t∈ [T0,T1]}and is, after scaling with factor h−2and shifting time T0to zero, A−1-close to a corresponding subset of the standard solution, o r (ii) the assertion (i) holds with T1replaced by some time t+∈(T0,T1), where t+is a surgery time; moreover, for each point in BT0(x0,Ah), the solution is deﬁned for t∈[T0,t+)but is not deﬁned past t+(i.e., the whole ball BT0(x0,Ah)is cut oﬀ at the time t+). Proof. LetQbe the maximum of the scalar curvature of the standard soluti on in the time interval [0 ,θ] and choose a large positive integer Nso that ∆t=(T1−T0) N< εη−1Q−1h2, where the positive constant ηis given in the canonical neighborhood assumption. Set tk=T0+k∆t,k= 0,1,...,N . From Lemma 7.3.4, the geodesic ball BT0(x0,A0h) at timeT0, withA0=δ−1 2 is, after scaling with factor h−2,δ1 2-close to the corresponding ball in the standard capped inﬁnite cylinder with the center near the tip. Assume ﬁrst that for each point inBT0(x0,A0h), the solution is deﬁned on [ T0,t1]. By the gradient estimates (7.3.4) in the canonical neighborhood assumption and the choice of ∆ twe have a uniform curvature bound on this set for h−2-scaled metric. Then by the uniqueness theorem (Theorem 1.2.4), if δ1 2→0 (i.e.A0=δ−1 2→+∞), the solution with h−2-scaled metric will converge to the standard solution in the C∞ loctopology. Therefore we can deﬁneA1, depending only on A0and tending to inﬁnity with A0, such that the solution in the parabolic region P(x0,T0,A1h,t1−T0)/defines{(x,t)|x∈Bt(x0,A1h),t∈ [T0,T0+ (t1−T0)]}is, after scaling with factor h−2and shifting time T0to zero, A−1 1-close to the corresponding subset in the standard solution . In particular, the scalar curvature on this subset does not exceed 2 Qh−2. Now if for each point in BT0(x0,A1h) the solution is deﬁned on [ T0,t2], then we can repeat the procedure, deﬁningA2, such that the solution in the parabolic region P(x0,T0,A2h,t2−T0)/defines {(x,t)|x∈Bt(x0,A2h),t∈[T0,T0+ (t2−T0)]}is, after scaling with factor h−2 and shifting time T0to zero,A−1 2-close to the corresponding subset in the standard THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 431 solution. Again, the scalar curvature on this subset still d oes not exceed 2 Qh−2. Continuing this way, we eventually deﬁne AN. Note that Nis depends only on θ andε. Thus there exists a positive ¯δ=¯δ(A,θ,ε ) such that for δ <¯δ, we have A0> A1>···> A N> A, and assertion (i) holds when the solution is deﬁned on BT0(x0,A(N−1)h)×[T0,T1]. The above argument shows that either assertion (i) holds, or there exists some k(0≤k≤N−1) and a surgery time t+∈(tk,tk+1] such that the solution on BT0(x0,Akh) is deﬁned on [ T0,t+), but for some point of this set it is not deﬁned past t+. Now we consider the latter case. Clearly the above argument also shows that the parabolic region P(x0,T0,Ak+1h,t+−T0)/defines{(x,t)|x∈Bt(x,A k+1h),t∈[T0,t+)} is, after scaling with factor h−2and shifting time T0to zero,A−1 k+1-close to the corresponding subset in the standard solution. In particul ar, as time tends to t+, the ballBT0(x0,Ak+1h) keeps on looking like a cap. Since the scalar curvature onBT0(x0,Akh)×[T0,tk] does not exceed 2 Qh−2, it follows from the pinching as- sumption, the gradient estimates in the canonical neighbor hood assumption and the evolution equation of the metric that the diameter of the set BT0(x0,Akh) at any timet∈[T0,t+) is bounded from above by 4 δ−1 2h. These imply that no point of the ballBT0(x0,Akh) at any time near t+can be the center of a δ-neck for any 0<δ< ¯δ(A,θ,ε ) with ¯δ(A,θ,ε )>0 small enough, since 4 δ−1 2his much smaller than δ−1h. However the solution disappears somewhere in the set BT0(x0,Akh) at time t+by a cutoﬀ surgery and the surgery is always done along the mid dle two-sphere of aδ-neck. So the set BT0(x0,Akh) at timet+is a part of a capped horn. (Recall that we have declared extinct every compact component with p ositive curvature and every compact component lying in Ω \Ωρ). Hence for each point of BT0(x0,Akh) the solution terminates at t+. This proves assertion (ii). Corollary 7.3.7 ( Perelman [104] ).For anyl <∞one can ﬁnd A=A(l)< ∞andθ=θ(l),0< θ < 1, with the following property. Suppose we are in the situation of the lemma above, with δ<¯δ(A,θ,ε ). Consider smooth curves γin the set BT0(x0,Ah), parametrized by t∈[T0,Tγ], such that γ(T0)∈BT0(x0,Ah 2)and either Tγ=T1<T, orTγ<T1andγ(Tγ)∈∂BT0(x0,Ah), wherex0is any ﬁxed point on a gluing cap at T0andT1=min{T,T0+θh2}. Then /integraldisplayTγ T0(R(γ(t),t) +|˙γ(t)|2)dt>l. Proof. We know from Proposition 7.3.3 that on the standard solution , /integraldisplayθ 0Rdt≥const./integraldisplayθ 0(1−t)−1dt =−const.·log(1−θ). By choosing θ=θ(l) suﬃciently close to 1 we have the desired estimate for the standard solution. Let us consider the ﬁrst case: Tγ=T1<T. Forθ=θ(l) ﬁxed above, by Lemma 7.3.6, our solution in the subset BT0(x0,Ah) and in the time interval [ T0,Tγ] is, after scaling with factor h−2and shifting time T0to zero,A−1-close to the corresponding 432 H.-D. CAO AND X.-P. ZHU subset in the standard solution for any suﬃciently large A. So we have /integraldisplayTγ T0(R(γ(t),t) +|˙γ(t)|2)dt≥const./integraldisplayθ 0(1−t)−1dt =−const.·log(1−θ). Hence we have obtained the desired estimate in the ﬁrst case. We now consider the second case: Tγ< T1andγ(Tγ)∈∂BT0(x0,Ah). Let θ=θ(l) be chosen above and let Q=Q(l) be the maximum of the scalar curvature on the standard solution in the time interval [0 ,θ]. On the standard solution, we can choose A=A(l) so large that for each t∈[0,θ], distt(x0,∂B0(x0,A))≥dist0(x0,∂B0(x0,A))−4(Q+ 1)t ≥A−4(Q+ 1)θ ≥4 5A and distt/parenleftbigg x0,∂B0/parenleftbigg x0,A 2/parenrightbigg/parenrightbigg ≤A 2, where we have used Lemma 3.4.1(ii) in the ﬁrst inequality. No w our solution in the subsetBT0(x0,Ah) and in the time interval [ T0,Tγ] is, after scaling with factor h−2 and shifting time T0to zero,A−1-close to the corresponding subset in the standard solution. This implies that for A=A(l) large enough 1 5Ah≤/integraldisplayTγ T0|˙γ(t)|dt≤/parenleftigg/integraldisplayTγ T0|˙γ(t)|2dt/parenrightigg1 2 ·(Tγ−T0)1 2, Hence /integraldisplayTγ T0(R(γ(t),t) +|˙γ(t)|2)dt≥A2 25θ>l. This proves the desired estimate. 7.4. Justiﬁcation of the Canonical Neighborhood Assumptio ns.We con- tinue the induction argument for the construction of a long- time solution to the Ricci ﬂow with surgery. Let us recall what we have done in the previo us section. Let εbe an arbitrarily given positive constant satisfying 0 <ε≤1/100. For an arbitrarily given compact orientable normalized three-manifold, we evolve i t by the Ricci ﬂow. We may assume that the solution goes singular at some time 0 <t+ 1<+∞and know that the solution satisﬁes the a priori assumptions (with accuracy ε) on [0,t+ 1) for a nonincreas- ing positive function r=r1(t) (deﬁned on [0 ,+∞)). Suppose that we have a solution to the Ricci ﬂow with surgery, deﬁned on [0 ,t+ k) with 0<t+ 1<t+ 2<···<t+ k<+∞, satisfying the a priori assumptions (with accuracy ε) for some nonincreasing positive functionr=rk(t) (deﬁned on [0 ,+∞)), going singular at time t+ kand having δi-cutoﬀ surgeries at each time t+ i, 1≤i≤k−1, whereδi<¯δ(t+ i) for each 1 ≤i≤k−1. Then for any 0<δk<¯δ(t+ k), we can perform δk-cutoﬀ surgeries at the time t+ kand extend the solution to the interval [0 ,t+ k+1) witht+ k+1>t+ k. Here ¯δ(t) is the positive function THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 433 deﬁned in (7.3.16). We have already shown in Lemma 7.3.4 that the extended solution still satisﬁes the pinching assumption on [0 ,t+ k+1). In view of Theorem 7.1.1, there always is a nonincreasing pos itive function r= rk+1(t), deﬁned on [0 ,+∞), such that the canonical neighborhood assumption (with accuracyε) holds on the extended time interval [0 ,t+ k+1) with the positive function r= rk+1(t). Nevertheless, in order to prevent the surgery times from a ccumulating, the key is to choose the nonincreasing positive functions r=ri(t),i= 1,2,..., uniformly. That is, to justify the canonical neighborhood assumption ( with accuracy ε) for the indeﬁnitely extending solution, we need to show that there e xists a nonincreasing positive function /tildewider(t), deﬁned on [0 ,+∞), which is independent of k, such that the above chosen nonincreasing positive functions satisfy ri(t)≥/tildewider(t),on [0,+∞), for alli= 1,2,...,k + 1. By a further restriction on the positive function ¯δ(t), we can verify this after proving the following assertion which was stated by Perelma n in [104]. Proposition 7.4.1 ( Justiﬁcation of the canonical neighborhood assumption ). Given any small ε >0, there exist decreasing sequences 0</tildewiderj< ε,κj>0, and 0</tildewideδj< ε2,j= 1,2,···, with the following property. Deﬁne the positive function /tildewideδ(t)on[0,+∞)by/tildewideδ(t) =/tildewideδjfort∈[(j−1)ε2,jε2). Suppose there is a surgically modiﬁed solution, deﬁned on [0,T)withT <+∞, to the Ricci ﬂow which satisﬁes the following: (1) it starts on a compact orientable three-manifold with no rmalized initial metric, and (2) it has only a ﬁnite number of surgeries such that each surg ery at a time t∈(0,T)is aδ(t)-cutoﬀ surgery with 0<δ(t)≤min{/tildewideδ(t),¯δ(t)}. Then on each time interval [(j−1)ε2,jε2]/intersectiontext[0,T),j= 1,2,···, the solution satisﬁes theκj-noncollapsing condition on all scales less than εand the canonical neighborhood assumption (with accuracy ε) withr=/tildewiderj. Here and in the following, we call a (three-dimensional) sur gically modiﬁed solu- tiongij(t),0≤t<T,κ-noncollapsed at (x0,t0) on the scales less than ρ(for some κ>0,ρ>0) if it satisﬁes the following property: whenever r<ρ and |Rm(x,t)| ≤r−2 for all those ( x,t)∈P(x0,t0,r,−r2) ={(x′,t′)|x′∈Bt′(x0,r),t′∈[t0−r2,t0]}, for which the solution is deﬁned, we have Volt0(Bt0(x0,r))≥κr3. Before we give the proof of the proposition, we need to verify aκ-noncollapsing estimate which was given by Perelman in [104]. Lemma 7.4.2. Given any 0< ε≤¯ε0(for some suﬃciently small universal constant ¯ε0),suppose we have constructed the sequences satisfying the pr oposition for 1≤j≤m(for some positive integer m). Then there exists κ>0, such that for any 434 H.-D. CAO AND X.-P. ZHU r,0< r < ε , one can ﬁnd /tildewideδ=/tildewideδ(r,ε),0</tildewideδ < ε2, which may also depend on the already constructed sequences, with the following propert y. Suppose we have a solution with a compact orientable normalized three-manifold as ini tial data, to the Ricci ﬂow with ﬁnite number of surgeries on a time interval [0,¯T]withmε2≤¯T <(m+ 1)ε2, satisfying the assumptions and the conclusions of Proposit ion7.4.1on[0,mε2), and the canonical neighborhood assumption (with accuracy ε)withron[mε2,¯T], as well as0< δ(t)≤min{/tildewideδ,¯δ(t)}for anyδ-cutoﬀ surgery with δ=δ(t)at a timet∈ [(m−1)ε2,¯T]. Then the solution is κ-noncollapsed on [0,¯T]for all scales less than ε. Proof. Consider a parabolic neighborhood P(x0,t0,r0,−r2 0)/defines{(x,t)|x∈Bt(x0,r0),t∈[t0−r2 0,t0]} withmε2≤t0≤¯Tand 0<r0<ε, where the solution satisﬁes |Rm| ≤r−2 0, whenever it is deﬁned. We will use an argument analogous to the proof of Theorem 3.3.2 (no local collapsing theorem I) to prove (7.4.1) Vol t0(Bt0(x0,r0))≥κr3 0. Letηbe the universal positive constant in the deﬁnition of the ca nonical neigh- borhood assumption. Without loss of generality, we always a ssumeη≥10. Firstly, we want to show that one may assume r0≥1 2ηr. Obviously, the curvature satisﬁes the estimate |Rm(x,t)| ≤20r−2 0, for those (x,t)∈P(x0,t0,1 2ηr0,−1 8ηr2 0) ={(x,t)|x∈Bt(x0,1 2ηr0),t∈[t0−1 8ηr2 0,t0]}, for which the solution is deﬁned. When r0<1 2ηr, we can enlarge r0to somer′ 0∈[r0,r] so that |Rm| ≤20r′−2 0 onP(x0,t0,1 2ηr′ 0,−1 8ηr′2 0) (whenever it is deﬁned), and either the equality holds some - where inP(x0,t0,1 2ηr′ 0,−(1 8ηr′2 0+ǫ′)) for any arbitrarily small ǫ′>0 orr′ 0=r. In the case that the equality holds somewhere, it follows fro m the pinching as- sumption that we have R>10r′−2 0 somewhere in P(x0,t0,1 2ηr′ 0,−(1 8ηr′2 0+ǫ′)) for any arbitrarily small ǫ′>0. Here, without loss of generality, we have assumed ris suitably small. Then by the gradient estimates in the deﬁnition of the canonical neighborhood as sumption, we know R(x0,t0)>r′−2 0≥r−2. Hence the desired noncollapsing estimate (7.4.1) in this ca se follows directly from the canonical neighborhood assumption. (Recall that we have ex cluded every compact component which has positive sectional curvature in the sur gery procedure and then we have excluded them from the list of canonical neighborhoo ds. Here we also used the standard volume comparison when the canonical neighbor hood is anε-cap.) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 435 While in the case that r′ 0=r, we have the curvature bound |Rm(x,t)| ≤/parenleftbigg1 2ηr/parenrightbigg−2 , for those ( x,t)∈P(x0,t0,1 2ηr,−(1 2ηr)2) ={(x,t)|x∈Bt(x0,1 2ηr),t∈[t0− (1 2ηr)2,t0]}, for which the solution is deﬁned. It follows from the standa rd volume comparison that we only need to verify the noncollapsing est imate (7.4.1) for r0=1 2ηr. Thus we have reduced the proof to the case r0≥1 2ηr. Recall from Theorem 3.3.2 that if a solution is smooth everyw here, we can get a lower bound for the volume of the ball Bt0(x0,r0) as follows: deﬁne τ(t) =t0−t and consider Perelman’s reduced volume function and the Li- Yau-Perelman distance associated to the point x0; take a point ¯ xat the time t=ε2so that the Li-Yau- Perelman distance lattains its minimum lmin(τ) =l(¯x,τ)≤3 2forτ=t0−ε2; use it to obtain an upper bound for the Li-Yau-Perelman distance fromx0to each point ofB0(¯x,1), thus getting a lower bound for Perelman’s reduced volume atτ=t0; apply the monotonicity of Perelman’s reduced volume to dedu ce a lower bound for Perelman’s reduced volume at τnear 0, and then get the desired estimate for the volume of the ball Bt0(x0,r0). Now since our solution has undergone surgeries, we need to localize this argument to the region which is unaﬀect ed by surgery. We call a space-time curve in the solution track admissible if it stays in the space- time region unaﬀected by surgery, and we call a space-time cu rve in the solution track abarely admissible curve if it is on the boundary of the set of admissible curves. First of all, we want to estimate the L-length of a barely admissible curve. Claim. For anyL <∞one can ﬁnd ¯δ=¯δ(L,r,/tildewiderm,ε)>0 with the following property. Suppose that we have a curve γ, parametrized by t∈[T0,t0], (m−1)ε2≤ T0<t0, such that γ(t0) =x0,T0is a surgery time, and γ(T0) lies in the gluing cap. Suppose also each δ-cutoﬀ surgery at a time in [( m−1)ε2,¯T] hasδ≤¯δ. Then we have an estimate (7.4.2)/integraldisplayt0 T0√t0−t(R+(γ(t),t) +|˙γ(t)|2)dt≥L whereR+= max {R,0}. Sincer0≥1 2ηrand|Rm| ≤r−2 0onP(x0,t0,r0,−r2 0) (whenever it is deﬁned), we can require ¯δ >0, depending on rand/tildewiderm, to be so small that γ(T0) does not lie in the regionP(x0,t0,r0,−r2 0). Let ∆tbe the maximal number such that γ|[t0−∆t,t0]⊂ P(x0,t0,r0,−∆t) (i.e.,t0−∆tis the ﬁrst time when γescapes the parabolic region P(x0,t0,r0,−r2 0)).Obviously we only need to consider the case: /integraldisplayt0 t0−∆t√t0−t(R+(γ(t),t) +|˙γ(t)|2)dt<L. We observe that ∆ tcan be bounded from below in terms of Landr0. Indeed, if ∆t≥r2 0, there is nothing to prove. Thus we may assume ∆ t<r2 0. By the curvature bound |Rm| ≤r−2 0onP(x0,t0,r0,−r2 0) and the Ricci ﬂow equation we see /integraldisplayt0 t0−∆t|˙γ(t)|dt≥cr0 436 H.-D. CAO AND X.-P. ZHU for some universal positive constant c. On the other hand, by the Cauchy-Schwarz inequality, we have /integraldisplayt0 t0−∆t|˙γ(t)|dt≤/parenleftbigg/integraldisplayt0 t0−∆t√t0−t(R++|˙γ|2)dt/parenrightbigg1 2 ·/parenleftbigg/integraldisplayt0 t0−∆t1√t0−tdt/parenrightbigg1 2 ≤(2L)1 2(∆t)1 4, which implies (7.4.3) (∆ t)1 2≥c2r2 0 2L. Thus /integraldisplayt0 T0√t0−t(R++|˙γ|2)dt≥/integraldisplayt0−∆t T0√t0−t(R++|˙γ|2)dt ≥(∆t)1 2/integraldisplayt0−∆t T0(R++|˙γ|2)dt ≥/parenleftbigg min/braceleftbiggc2r2 0 2L,r0/bracerightbigg/parenrightbigg/integraldisplayt0−∆t T0(R++|˙γ|2)dt, while by Corollary 7.3.7, we can ﬁnd ¯δ=¯δ(L,r,/tildewiderm,ε)>0 so small that /integraldisplayt0−∆t T0(R++|˙γ|2)dt≥L/parenleftbigg min/braceleftbiggc2r2 0 2L,r0/bracerightbigg/parenrightbigg−1 . Then we have proved the desired assertion (7.4.2). Recall that for a curve γ, parametrized by τ=t0−t∈[0,¯τ],withγ(0) =x0and ¯τ≤t0−(m−1)ε2, we haveL(γ) =/integraltext¯τ 0√τ(R+|˙γ|2)dτ. We can also deﬁne L+(γ) by replacing RwithR+in the previous formula. Recall that R≥ −1 at the initial timet= 0 for the normalized initial manifold. Recall that the surg eries occur at the parts where the scalar curvatures are very large. Thus we can apply the maximum principle to conclude that the solution with surgery still s atisﬁesR≥ −1 everywhere in space-time. This implies (7.4.4) L+(γ)≤L(γ) + (2ε2)3 2. By applying the assertion (7.4.2), we now choose ˜δ >0 (depending on r,εand/tildewiderm) such that as each δ-cutoﬀ surgery at the time interval t∈[(m−1)ε2,T] hasδ≤˜δ, every barely admissible curve γfrom (x0,t0) to a point ( x,t) (witht∈[(m−1)ε2,t0)) has L+(γ)≥22√ 2. Thus if the Li-Yau-Perelman distance from ( x0,t0) to a point ( x,t) (witht∈[(m− 1)ε2,t0)) is achieved by a space-time curve which is not admissible, then its Li-Yau- Perelman distance has (7.4.5) l≥L+−(2ε2)3 2 2√ 2ε>10ε−1. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 437 We also observe that the absolute value of l(x0,τ) is very small as τcloses to zere. Thus the maximum principle argument in Corollary 3.2.6 stil l works for our solutions with surgery because barely admissible curves do not attain the minimum. So we conclude that lmin(¯τ) = min {l(x,¯τ)|xlies in the solution manifold at t0−¯τ} ≤3 2 for ¯τ∈(0,t0−(m−1)ε2]. In particular, there exists a minimizing curve γoflmin(t0− (m−1)ε2), deﬁned on τ∈[0,t0−(m−1)ε2] withγ(0) =x0, such that L+(γ)≤3 2·2√ 2ε+ 2√ 2ε3(7.4.6) ≤5ε, since 0<ε≤¯ε0with ¯ε0suﬃciently small (to be further determined). Consequently , there exists a point (¯ x,¯t) on the minimizing curve γwith¯t∈[(m−1)ε2+1 4ε2,(m− 1)ε2+3 4ε2] (i.e.,τ∈[t0−(m−1)ε2−3 4ε2,t0−(m−1)ε2−1 4ε2]) such that (7.4.7) R(¯x,¯t)≤25/tildewider−2 m. Otherwise, we have L+(γ)≥/integraldisplayt0−(m−1)ε2−1 4ε2 t0−(m−1)ε2−3 4ε2√τR(γ(τ),t0−τ)dτ >25/tildewider−2 m/radicalbigg 1 4ε2/parenleftbigg1 2ε2/parenrightbigg >5ε, since 0</tildewiderm<ε. This contradicts (7.4.6). Next we want to get a lower bound for Perelman’s reduced volum e of a ball around ¯xof radius about /tildewidermat some time slightly before ¯t. Denote byθ1=1 16η−1andθ2=1 64η−1, whereηis the universal positive constant in the gradient estimates (7.3.4). Since the solution satis ﬁes the canonical neighbor- hood assumption on the time interval [( m−1)ε2,mε2), it follows from the gradient estimates (7.3.4) that (7.4.8) R(x,t)≤400/tildewider−2 m for those (x,t)∈P(¯x,¯t,θ1/tildewiderm,−θ2/tildewider2 m)/defines{(x′,t′)|x′∈Bt′(¯x,θ1/tildewiderm),t′∈[¯t−θ2/tildewider2 m,¯t]}, for which the solution is deﬁned. And since the scalar curvat ure at the points where theδ-cutoﬀ surgeries occur in the time interval [( m−1)ε2,mε2) is at least (/tildewideδ)−2/tildewider−2 m, the solution is well-deﬁned on the whole parabolic region P(¯x,¯t,θ1/tildewiderm,−θ2/tildewider2 m) (i.e., this parabolic region is unaﬀected by surgery). Thus by comb ining (7.4.6) and (7.4.8), we know that the Li-Yau-Perelman distance from ( x0,t0) to each point of the ball B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm) is uniformly bounded by some universal constant. Let us deﬁ ne Perelman’s reduced volume of the ball B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm), by /tildewideVt0−¯t+θ2 /CTr2 m(B¯t−θ2 /CTr2 m(¯x,θ1/tildewiderm)) =/integraldisplay B¯t−θ2 /CTr2m(¯x,θ1 /CTrm)(4π(t0−¯t+θ2/tildewider2 m))−3 2 ·exp(−l(q,t0−¯t+θ2/tildewider2 m))dV¯t−θ2 /CTr2m(q), 438 H.-D. CAO AND X.-P. ZHU wherel(q,τ) is the Li-Yau-Perelman distance from ( x0,t0). Hence by the κm- noncollapsing assumption on the time interval [( m−1)ε2,mε2), we conclude that Perelman’s reduced volume of the ball B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm) is bounded from below by a positive constant depending only on κmand/tildewiderm. Finally we want to apply a local version of the monotonicity o f Perelman’s reduced volume to get a lower bound estimate for the volume of the ball Bt0(x0,r0). We have seen that the Li-Yau-Perelman distance from ( x0,t0) to each point of the ballB¯t−θ2 /CTr2 m(¯x,θ1/tildewiderm) is uniformly bounded by some universal constant. Now we can choose a suﬃciently small (universal) positive constant ¯ ε0such that when 0 <ε≤¯ε0, by (7.4.5), all the points in the ball B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm) can be connected to ( x0,t0) by shortest L-geodesics, and all of these L-geodesics are admissible (i.e., they stay in the region unaﬀected by surgery). The union of all shortest L-geodesics from ( x0,t0) to the ball B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm) deﬁned by CB¯t−θ2 /CTr2m(¯x,θ1/tildewiderm) ={(x,t)|(x,t) lies in a shortest L-geodesic from ( x0,t0) to a point in B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm)}, forms a cone- like subset in space-time with the vertex ( x0,t0). DenoteB(t) by the intersection of the cone-like subset CB¯t−θ2 /CTr2 m(¯x,θ1/tildewiderm) with the time-slice at t. Perelman’s reduced volume of the subset B(t) is given by /tildewideVt0−t(B(t)) =/integraldisplay B(t)(4π(t0−t))−3 2exp(−l(q,t0−t))dVt(q). Since the cone-like subset CB¯t−θ2 /CTr2m(¯x,θ1/tildewiderm) lies entirely in the region unaﬀected by surgery, we can apply Perelman’s Jacobian comparison theor em (Theorem 3.2.7) to conclude that /tildewideVt0−t(B(t))≥/tildewideVt0−¯t+θ2 /CTr2m(B¯t−θ2 /CTr2m(¯x,θ1/tildewiderm)) (7.4.9) ≥c(κm,/tildewiderm), for allt∈[¯t−θ2/tildewider2 m,t0], wherec(κm,/tildewiderm) is some positive constant depending only on κmand/tildewiderm. Setξ=r−1 0Volt0(Bt0(x0,r0))1 3. Our purpose is to give a positive lower bound forξ. Without loss of generality, we may assume ξ<1 4, thus 0<ξr2 0<t0−¯t+θ2/tildewider2 m. Denote by/tildewideB(t0−ξr2 0) the subset of the time-slice {t=t0−ξr2 0}of which every point can be connected to ( x0,t0) by an admissible shortest L-geodesic. Clearly, B(t0−ξr2 0)⊂/tildewideB(t0−ξr2 0). We now argue as in the proof of Theorem 3.3.2 to bound Perelman’s reduced volume of /tildewideB(t0−ξr2 0) from above. Sincer0≥1 2ηrand˜δ=˜δ(r,ε,/tildewiderm) suﬃciently small, the whole region P(x0,t0,r0, −r2 0) is unaﬀected by surgery. Then by exactly the same argument a s in deriving (3.3.5), we see that there exists a universal positive const antξ0such that when 0 < ξ≤ξ0, there holds (7.4.10) Lexp{|υ|≤1 4ξ−1 2}(ξr2 0)⊂Bt0(x0,r0). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 439 Perelman’s reduced volume of /tildewideB(t0−ξr2 0) is given by /tildewideVξr2 0(/tildewideB(t0−ξr2 0)) (7.4.11) =/integraldisplay/CTB(t0−ξr2 0)(4πξr2 0)−3 2exp(−l(q,ξr2 0))dVt0−ξr2 0(q) =/integraldisplay/CTB(t0−ξr2 0)∩Lexp {|υ|≤1 4ξ−1 2}(ξr2 0)(4πξr2 0)−3 2exp(−l(q,ξr2 0))dVt0−ξr2 0(q) +/integraldisplay/CTB(t0−ξr2 0)\Lexp {|υ|≤1 4ξ−1 2}(ξr2 0)(4πξr2 0)−3 2exp(−l(q,ξr2 0))dVt0−ξr2 0(q). The ﬁrst term on the RHS of (7.4.11) can be estimated by /integraldisplay/CTB(t0−ξr2 0)∩Lexp {|υ|≤1 4ξ−1 2}(ξr2 0)(4πξr2 0)−3 2exp(−l(q,ξr2 0))dVt0−ξr2 0(q) (7.4.12) ≤eCξ(4π)−3 2·ξ3 2 for some universal constant C, as in deriving (3.3.7). While as in deriving (3.3.8), the second term on the RHS of (7.4.11) can be estimated by /integraldisplay/CTB(t0−ξr2 0)\Lexp {|υ|≤1 4ξ−1 2}(ξr2 0)(4πξr2 0)−3 2exp(−l(q,ξr2 0))dVt0−ξr2 0(q) (7.4.13) ≤/integraldisplay {|υ|>1 4ξ−1 2}(4πτ)−3 2exp(−l(τ))J(τ)|τ=0dυ = (4π)−3 2/integraldisplay {|υ|>1 4ξ−1 2}exp(−|υ|2)dυ, where we have used Perelman’s Jacobian comparison theorem ( Theorem 3.2.7) in the ﬁrst inequality. Hence the combination of (7.4.9), (7.4 .11), (7.4.12) and (7.4.13) boundsξfrom below by a positive constant depending only on κmand/tildewiderm. Therefore we have completed the proof of the lemma. Now we can prove the proposition. Proof of Proposition 7.4.1. The proof of the proposition is by induction: having constructed our sequences for 1 ≤j≤m, we make one more step, deﬁning /tildewiderm+1, κm+1,/tildewideδm+1, and redeﬁning /tildewideδm=/tildewideδm+1. In view of the previous lemma, we only need to deﬁne/tildewiderm+1and/tildewideδm+1. In Theorem 7.1.1 we have obtained the canonical neighborhoo d structure for smooth solutions. When adapting the arguments in the proof o f Theorem 7.1.1 to the present surgical solutions, we will encounter the new di ﬃculty of how to take a limit for the surgically modiﬁed solutions. The idea to over come the diﬃculty con- sists of two parts. The ﬁrst part, due to Perelman [104], is to choose/tildewideδmand/tildewideδm+1 small enough to push the surgical regions to inﬁnity in space . (This is the reason why we need to redeﬁne /tildewideδm=/tildewideδm+1.) The second part is to show that solutions are smooth on some small, but uniform, time intervals (on compac t subsets) so that we can apply Hamilton’s compactness theorem, since we only hav e curvature bounds; 440 H.-D. CAO AND X.-P. ZHU otherwise Shi’s interior derivative estimate may not be app licable. In fact, the sec- ond part is more crucial. That is just concerned with the ques tion of whether the surgery times accumulate or not. Our argument will use the ca nonical neighborhood characterization of the standard solution in Lemma 7.3.5. We now start to prove the proposition by contradiction. Supp ose for sequence of positive numbers rαand/tildewideδαβ, satisfying rα→0 asα→ ∞ and/tildewideδαβ≤1 α·β(→0), there exist sequences of solutions gαβ ijto the Ricci ﬂow with surgery, where each of them has only a ﬁnite number of cutoﬀ surgeries and has a compact or ientable normalized three-manifold as initial data, so that the following two as sertions hold: (i) eachδ-cutoﬀ at a time t∈[(m−1)ε2,(m+ 1)ε2] satisﬁesδ≤/tildewideδαβ; and (ii) the solutions satisfy the statement of the proposition on [0,mε2], but violate the canonical neighborhood assumption (with accuracy ε) withr=rαon [mε2,(m+ 1)ε2]. For each solution gαβ ij, we choose ¯t(depending on αandβ) to be the nearly ﬁrst time for which the canonical neighborhood assumption (with accuracyε) is violated. More precisely, we choose ¯t∈[mε2,(m+ 1)ε2] so that the canonical neighborhood assumption with r=rαand with accuracy parameter εis violated at some (¯ x,¯t), however the canonical neighborhood assumption with accura cy parameter 2 εholds ont∈[mε2,¯t]. After passing to subsequences, we may assume each /tildewideδαβis less than the/tildewideδin Lemma 7.4.2 with r=rαwhenαis ﬁxed. Then by Lemma 7.4.2 we have uniformκ-noncollapsing on all scales less than εon [0,¯t] with some κ>0 independent ofα,β. Slightly abusing notation, we will often drop the indices αandβ. Let/tildewidegαβ ijbe the rescaled solutions around (¯ x,¯t) with factors R(¯x,¯t)(≥r−2→+∞) and shift the times ¯tto zero. We hope to take a limit of the rescaled solutions for subsequences of α,β→ ∞ and show the limit is an orientable ancient κ-solution, which will give the desired contradiction. We divide our arg uments into the following six steps. Step1. Let (y,ˆt) be a point on the rescaled solution /tildewidegαβ ijwith/tildewideR(y,ˆt)≤A(for someA≥1) and ˆt∈[−(¯t−(m−1)ε2)R(¯x,¯t),0]. Then we have estimate (7.4.14) /tildewideR(x,t)≤10A for those ( x,t) in the parabolic neighborhood P(y,ˆt,1 2η−1A−1 2,−1 8η−1A−1)/defines {(x′,t′)|x′∈/tildewideBt′(y,1 2η−1A−1 2),t′∈[ˆt−1 8η−1A−1,ˆt]}, for which the rescaled so- lution is deﬁned. Indeed, as in the ﬁrst step of the proof of Theorem 7.1.1, this follows directly from the gradient estimates (7.3.4) in the canonical neighb orhood assumption with parameter 2 ε. Step2. In this step, we will prove three time extension results. Assertion 1. For arbitrarily ﬁxed α, 0< A < +∞, 1≤C < +∞and 0≤B <1 2ε2(rα)−2−1 8η−1C−1, there is a β0=β0(ε,A,B,C ) (independent of α) such that if β≥β0and the rescaled solution /tildewidegαβ ijon the ball/tildewideB0(¯x,A) is deﬁned on a time interval [ −b,0] with 0 ≤b≤Band the scalar curvature satisﬁes /tildewideR(x,t)≤C,on/tildewideB0(¯x,A)×[−b,0], then the rescaled solution /tildewidegαβ ijon the ball/tildewideB0(¯x,A) is also deﬁned on the extended time interval [ −b−1 8η−1C−1,0]. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 441 Before giving the proof, we make a simple observation : once a space point in the Ricci ﬂow with surgery is removed by surgery at some time, then it never appears for later time; if a space point at some time tcannot be deﬁned before the time t, then either the point lies in a gluing cap of the surgery at tim etor the time tis the initial time of the Ricci ﬂow. Proof of Assertion 1. Firstly we claim that there exists β0=β0(ε,A,B,C ) such that whenβ≥β0, the rescaled solution /tildewidegαβ ijon the ball/tildewideB0(¯x,A) can be deﬁned before the time −b(i.e., there are no surgeries interfering in /tildewideB0(¯x,A)×[−b−ǫ′,−b] for some ǫ′>0). We argue by contradiction. Suppose not, then there is some po int ˜x∈/tildewideB0(¯x,A) such that the rescaled solution /tildewidegαβ ijat ˜xcannot be deﬁned before the time −b. By the above observation, there is a surgery at the time −bsuch that the point ˜ xlies in the instant gluing cap. Let˜h(=R(¯x,¯t)1 2h) be the cut-oﬀ radius at the time −bfor the rescaled solution. Clearly, there is a universal constant Dsuch thatD−1˜h≤/tildewideR(˜x,−b)−1 2≤D˜h. By Lemma 7.3.4 and looking at the rescaled solution at the tim e−b, the gluing cap and the adjacent δ-neck, of radius ˜h, constitute a ( /tildewideδαβ)1 2-capK. For any ﬁxed small positive constant δ′(much smaller than ε), we see that /tildewideB(−b)(˜x,(δ′)−1/tildewideR(˜x,−b)−1 2)⊂ K whenβlarge enough. We ﬁrst verify the following Claim 1. For any small constants 0 <˜θ<1,δ′>0, there exists a β(δ′,ε,˜θ)>0 such that when β≥β(δ′,ε,˜θ), we have (i) the rescaled solution /tildewidegαβ ijover/tildewideB(−b)(˜x,(δ′)−1˜h) is deﬁned on the time interval [−b,0]∩[−b,−b+ (1−˜θ)˜h2]; (ii) the ball/tildewideB(−b)(˜x,(δ′)−1˜h) in the (/tildewideδαβ)1 2-capKevolved by the Ricci ﬂow on the time interval [ −b,0]∩[−b,−b+(1−˜θ)˜h2] is, after scaling with factor ˜h−2, δ′-close (in the C[δ′−1]topology) to the corresponding subset of the standard solution. This claim essentially follows from Lemma 7.3.6. Indeed, su ppose there is a surgery at some time˜˜t∈[−b,0]∩(−b,−b+ (1−˜θ)˜h2] which removes some point ˜˜x∈/tildewideB(−b)(˜x,(δ′)−1˜h). We assume˜˜t∈(−b,0] is the ﬁrst time with that property. Then by Lemma 7.3.6, there is a ¯δ=¯δ(δ′,ε,˜θ) such that if /tildewideδαβ<¯δ, then the ball /tildewideB(−b)(˜x,(δ′)−1˜h) in the (/tildewideδαβ)1 2-capKevolved by the Ricci ﬂow on the time interval [−b,˜˜t) is, after scaling with factor ˜h−2,δ′-close to the corresponding subset of the standard solution. Note that the metrics for times in [ −b,˜˜t) on/tildewideB(−b)(˜x,(δ′)−1˜h) are equivalent. By Lemma 7.3.6, the solution on /tildewideB(−b)(˜x,(δ′)−1˜h) keeps looking like a cap fort∈[−b,˜˜t). On the other hand, by the deﬁnition, the surgery is always d one along the middle two-sphere of a δ-neck with δ </tildewideδαβ. Then for βlarge, all the points in /tildewideB(−b)(˜x,(δ′)−1˜h) are removed (as a part of a capped horn) at the time˜˜t. But ˜x(near the tip of the cap) exists past the time˜˜t. This is a contradiction. Hence we have proved that/tildewideB(−b)(˜x,(δ′)−1˜h) is deﬁned on the time interval [ −b,0]∩[−b,−b+ (1−˜θ)˜h2]. 442 H.-D. CAO AND X.-P. ZHU Theδ′-closeness of the solution on /tildewideB(−b)(˜x,(δ′)−1h)×([−b,0]∩[−b,−b+(1−˜θ)˜h2]) with the corresponding subset of the standard solution foll ows from Lemma 7.3.6. Then we have proved Claim 1. We next verify the following Claim 2. There is ˜θ=˜θ(CB), 0<˜θ<1, such that b≤(1−˜θ)˜h2whenβlarge. Note from Proposition 7.3.3, there is a universal constant D′>0 such that the standard solution satisﬁes the following curvature estima te R(y,s)≥2D′ 1−s. We choose ˜θ=D′/2(D′+CB). Then for βlarge enough, the rescaled solution satisﬁes (7.4.15) /tildewideR(x,t)≥D′ 1−(t+b)˜h−2˜h−2 on/tildewideB(−b)(˜x,(δ′)−1˜h)×([−b,0]∩[−b,−b+ (1−˜θ)˜h2]). Supposeb≥(1−˜θ)˜h2. Then by combining with the assumption /tildewideR(˜x,t)≤Cfor t= (1−˜θ)˜h2−b, we have C≥D′ 1−(t+b)˜h−2˜h−2, and then 1≥(1−˜θ)/parenleftbigg 1 +D′ CB/parenrightbigg . This is a contradiction. Hence we have proved Claim 2. The combination of the above two claims shows that there is a p ositive constant 0<˜θ=˜θ(CB)<1 such that for any small δ′>0, there is a positive β(δ′,ε,˜θ) such that whenβ≥β(δ′,ε,˜θ), we haveb≤(1−˜θ)˜h2and the rescaled solution in the ball /tildewideB(−b)(˜x,(δ′)−1˜h) on the time interval [ −b,0] is, after scaling with factor ˜h−2,δ′-close ( in theC[(δ′)−1]topology) to the corresponding subset of the standard solut ion. By (7.4.15) and the assumption /tildewideR≤Con/tildewideB0(¯x,A)×[−b,0],we know that the cut-oﬀ radius ˜hat the time −bfor the rescaled solution satisﬁes ˜h≥/radicalbigg D′ C. Letδ′>0 be much smaller than εand min {A−1,A}. Since ˜d0(˜x,¯x)≤A, it follows that there is constant C(˜θ) depending only on ˜θsuch that ˜d(−b)(˜x,¯x)≤C(˜θ)A≪ (δ′)−1˜h. We now apply Lemma 7.3.5 with the accuracy parameter ε/2. LetC(ε/2) be the positive constant in Lemma 7.3.5. Without loss of gene rality, we may assume the positive constant C1(ε) in the canonical neighborhood assumption is larger than 4C(ε/2). Whenδ′(>0) is much smaller than εand min {A−1,A}, the point ¯ xat the time¯thas a neighborhood which is either a3 4ε-cap or a3 4ε-neck. Since the canonical neighborhood assumption with accuracy parameterεis vio- lated at (¯x,¯t), the neighborhood of the point ¯ xat the new time zero for the rescaled THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 443 solution must be a3 4ε-neck. By Lemma 7.3.5 (b), we know the neighborhood is the slice at the time zero of the parabolic neighborhood P(¯x,0,4 3ε−1/tildewideR(¯x,0)−1 2,−min{/tildewideR(¯x,0)−1,b}) (with/tildewideR(¯x,0) = 1) which is3 4ε-close (in the C[4 3ε−1]topology) to the corresponding subset of the evolving standard cylinder S2×Rover the time interval [ −min{b,1},0] with scalar curvature 1 at the time zero. If b≥1, the3 4ε-neck is strong, which is a contradiction. While if b<1, the3 4ε-neck at time −bis contained in the union of the gluing cap and the adjacent δ-neck where the δ-cutoﬀ surgery took place. Since εis small (sayε<1/100), it is clear that the point ¯ xat time −bis the center of an ε-neck which is entirely contained in the adjacent δ-neck. By the proof of Lemma 7.3.2, the adjacentδ-neck approximates an ancient κ-solution. This implies the point ¯ xat the time¯thas a strong ε-neck, which is also a contradiction. Hence we have proved that there exists β0=β0(ε,A,B,C ) such that when β≥β0, the rescaled solution on the ball /tildewideB0(¯x,A) can be deﬁned before the time −b. Let [tαβ A,0]⊃[−b,0] be the largest time interval so that the rescaled solution /tildewidegαβ ij can be deﬁned on /tildewideB0(¯x,A)×[tαβ A,0]. We ﬁnally claim that tαβ A≤ −b−1 8η−1C−1for βlarge enough. Indeed, suppose not, by the gradient estimates as in Step 1, w e have the curvature estimate /tildewideR(x,t)≤10C on/tildewideB0(¯x,A)×[tαβ A,−b]. Hence we have the curvature estimate /tildewideR(x,t)≤10C on/tildewideB0(¯x,A)×[tαβ A,0]. By the above argument there is a β0=β0(ε,A,B + 1 8η−1C−1,10C) such that for β≥β0, the solution in the ball /tildewideB0(¯x,A) can be de- ﬁned before the time tαβ A. This is a contradiction. Therefore we have proved Assertion 1. Assertion 2. For arbitrarily ﬁxed α, 0< A < +∞, 1≤C < +∞and 0< B <1 2ε2(rα)−2−1 50η−1, there is a β0=β0(ε,A,B,C ) (independent of α) such that ifβ≥β0and the rescaled solution /tildewidegαβ ijon the ball/tildewideB0(¯x,A) is deﬁned on a time interval [ −b+ǫ′,0] with 0< b≤Band 0< ǫ′<1 50η−1and the scalar curvature satisﬁes /tildewideR(x,t)≤Con/tildewideB0(¯x,A)×[−b+ǫ′,0], and there is a point y∈/tildewideB0(¯x,A) such that/tildewideR(y,−b+ǫ′)≤3 2, then the rescaled solution/tildewidegαβ ijatyis also deﬁned on the extended time interval [ −b−1 50η−1,0] and satisﬁes the estimate /tildewideR(y,t)≤15 fort∈[−b−1 50η−1,−b+ǫ′]. Proof of Assertion 2. We imitate the proof of Assertion 1. If the rescaled solution /tildewidegαβ ijatycannot be deﬁned for some time in [ −b−1 50η−1,−b+ǫ′), then there is a 444 H.-D. CAO AND X.-P. ZHU surgery at some time˜˜t∈[−b−1 50η−1,−b+ǫ′] such that ylies in the instant gluing cap. Let ˜h(=R(¯x,¯t)1 2h) be the cutoﬀ radius at the time˜˜tfor the rescaled solution. Clearly, there is a universal constant D >1 such that D−1˜h≤/tildewideR(y,˜˜t)−1 2≤D˜h. By the gradient estimates as in Step 1, the cutoﬀ radius satisﬁe s ˜h≥D−115−1 2. As in Claim 1 (i) in the proof of Assertion 1, for any small cons tants 0<˜θ<1 2, δ′>0, there exists a β(δ′,ε,˜θ)>0 such that for β≥β(δ′,ε,˜θ), there is no surgery interfering in /tildewideB˜˜t(y,(δ′)−1˜h)×([˜˜t,(1−˜θ)˜h2+˜˜t]∩(˜˜t,0]). Without loss of generality, we may assume that the universal constant ηis much larger than D. Then we have (1−˜θ)˜h2+˜˜t >−b+1 50η−1. As in Claim 2 in the proof of Assertion 1, we can use the curvature bound assumption to choose ˜θ=˜θ(B,C) such that (1 −˜θ)˜h2+˜˜t≥0; otherwise C≥D′ ˜θ˜h2 for some universal constant D′>1, and |˜˜t+b| ≤1 50η−1, which implies 1≥(1−˜θ)/parenleftigg 1 +D′ C/parenleftbig B+1 50η−1/parenrightbig/parenrightigg . This is a contradiction if we choose ˜θ=D′/2(D′+C(B+1 50η−1)). So there is a positive constant 0 <˜θ=˜θ(B,C)<1 such that for any δ′>0, there is a positive β(δ′,ε,˜θ) such that when β≥β(δ′,ε,˜θ), we have −˜˜t≤(1−˜θ)˜h2 and the solution in the ball /tildewideB˜˜t(˜x,(δ′)−1˜h) on the time interval [˜˜t,0] is, after scaling with factor ˜h−2,δ′-close (in the C[δ′−1]topology) to the corresponding subset of the standard solution. Then exactly as in the proof of Assertion 1, by using the canon ical neighborhood structure of the standard solution in Lemma 7.3.5, this give s the desired contradic- tion with the hypothesis that the canonical neighborhood as sumption with accuracy parameterεis violated at (¯ x,¯t), forβsuﬃciently large. The curvature estimate at the point yfollows from Step 1. Therefore the proof of Assertion 2 is complete. Note that the standard solution satisﬁes R(x1,t)≤D′′R(x2,t) for anyt∈[0,1 2] and any two points x1,x2, whereD′′≥1 is a universal constant. Assertion 3. For arbitrarily ﬁxed α, 0< A < +∞, 1≤C <+∞, there is a β0=β0(ε,AC1 2) such that if any point ( y0,t0) with 0 ≤ −t0<1 2ε2(rα)−2−1 8η−1C−1 of the rescaled solution /tildewidegαβ ijforβ≥β0satisﬁes/tildewideR(y0,t0)≤C, then either the rescaled solution at y0can be deﬁned at least on [ t0−1 16η−1C−1,t0] and the rescaled scalar curvature satisﬁes /tildewideR(y0,t)≤10Cfort∈/bracketleftig t0−1 16η−1C−1,t0/bracketrightig , THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 445 or we have /tildewideR(x1,t0)≤2D′′/tildewideR(x2,t0) for any two points x1,x2∈/tildewideBt0(y0,A), whereD′′is the above universal constant. Proof of Assertion 3. Suppose the rescaled solution /tildewidegαβ ijaty0cannot be deﬁned for somet∈[t0−1 16η−1C−1,t0); then there is a surgery at some time ˜t∈[t0− 1 16η−1C−1,t0] such that y0lies in the instant gluing cap. Let ˜h(=R(¯x,¯t)1 2h) be the cutoﬀ radius at the time ˜tfor the rescaled solution /tildewidegαβ ij. By the gradient estimates as in Step 1, the cutoﬀ radius satisﬁes ˜h≥D−110−1 2C−1 2, whereDis the universal constant in the proof of the Assertion 1. Sin ce we assume η is suitably larger than Das before, we have1 2˜h2+˜t > t 0. As in Claim 1 (ii) in the proof of Assertion 1, for arbitrarily small δ′>0, we know that for βlarge enough the rescaled solution on /tildewideB˜t(y0,(δ′)−1˜h)×[˜t,t0] is, after scaling with factor ˜h−2,δ′-close (in theC[(δ′)−1]topology) to the corresponding subset of the standard solut ion. Since (δ′)−1˜h≫Aforβlarge enough, Assertion 3 follows from the curvature estima te of standard solution in the time interval [0 ,1 2]. Step3. For any subsequence ( αk,βk) of (α,β) withrαk→0 andδαkβk→0 ask→ ∞, we next argue as in the second step of the proof of Theorem 7.1 .1 to show that the curvatures of the rescaled solutions ˜ gαkβk ijat the new times zero (after shifting) stay uniformly bounded at bounded distances from ¯xfor all suﬃciently large k. More precisely, we will prove the following assertion: Assertion 4. Given any subsequence of the rescaled solutions ˜ gαkβk ijwithrαk→ 0 andδαkβk→0 ask→ ∞, then for any L >0, there are constants C(L)>0 and k(L) such that the rescaled solutions ˜ gαkβk ijsatisfy (i)˜R(x,0)≤C(L) for all points xwith˜d0(x,¯x)≤Land allk≥1; (ii) the rescaled solutions over the ball ˜B0(¯x,L) are deﬁned at least on the time interval [ −1 16η−1C(L)−1,0] for allk≥k(L). Proof of Assertion 4. For eachρ>0, set M(ρ) = sup/braceleftig ˜R(x,0)|k≥1 and ˜d0(x,¯x)≤ρin the rescaled solutions ˜ gαkβk ij/bracerightig and ρ0= sup{ρ>0|M(ρ)<+∞}. Note that the estimate (7.4.14) implies that ρ0>0. For (i), it suﬃces to prove ρ0= +∞. We argue by contradiction. Suppose ρ0<+∞. Then there is a sequence of pointsyin the rescaled solutions ˜ gαkβk ijwith˜d0(¯x,y)→ρ0<+∞and˜R(y,0)→+∞. Denote byγa minimizing geodesic segment from ¯ xtoyand denote by ˜B0(¯x,ρ0) the open geodesic ball centered at ¯ xof radiusρ0on the rescaled solution ˜ gαkβk ij. First, we claim that for any 0 < ρ < ρ 0withρnearρ0, the rescaled solutions on the balls ˜B0(¯x,ρ) are deﬁned on the time interval [ −1 16η−1M(ρ)−1,0] for all large 446 H.-D. CAO AND X.-P. ZHU k. Indeed, this follows from Assertion 3 or Assertion 1. For th e later purpose in Step 6, we now present an argument by using Assertion 3. If the claim is not true, then there is a surgery at some time ˜t∈[−1 16η−1M(ρ)−1,0] such that some point ˜y∈˜B0(¯x,ρ) lies in the instant gluing cap. We can choose suﬃciently sma llδ′>0 such that 2 ρ0<(δ′)−1 2˜h, where ˜h≥D−120−1 2M(ρ)−1 2is the cutoﬀ radius of the rescaled solutions at ˜t. By applying Assertion 3 with (˜ y,0) = (y0,t0), we see that there is ak(ρ0,M(ρ))>0 such that when k≥k(ρ0,M(ρ)), /tildewideR(x,0)≤2D′′ for allx∈/tildewideB0(¯x,ρ). This is a contradiction as ρ→ρ0. Since for each ﬁxed 0 <ρ<ρ 0withρnearρ0, the rescaled solutions are deﬁned on ˜B0(¯x,ρ)×[−1 16η−1M(ρ)−1,0] for all large k, by Step 1 and Shi’s derivative estimate, we know that the covariant derivatives and higher order deri vatives of the curvatures on˜B0(¯x,ρ−(ρ0−ρ) 2)×[−1 32η−1M(ρ)−1,0] are also uniformly bounded. By the uniform κ-noncollapsing property and Hamilton’s compactness theor em (Theorem 4.1.5), after passing to a subsequence, we can assu me that the marked se- quence ( ˜B0(¯x,ρ0),/tildewidegαkβk ij,¯x) converges in the C∞ loctopology to a marked (noncomplete) manifold (B∞,/tildewideg∞ ij,¯x) and the geodesic segments γconverge to a geodesic segment (missing an endpoint) γ∞⊂B∞emanating from ¯ x. Clearly, the limit has nonnegative sectional curvature by t he pinching assumption. Consider a tubular neighborhood along γ∞deﬁned by V=/uniondisplay q0∈γ∞B∞(q0,4π(/tildewideR∞(q0))−1 2), where/tildewideR∞denotes the scalar curvature of the limit and B∞(q0,4π(/tildewideR∞(q0))−1 2) is the ball centered at q0∈B∞with the radius 4 π(/tildewideR∞(q0))−1 2. Let ¯B∞denote the completion of ( B∞,/tildewideg∞ ij), andy∞∈¯B∞the limit point of γ∞. Exactly as in the second step of the proof of Theorem 7.1.1, it follows from the canonical neighborhood assumption with accuracy parameter 2 εthat the limiting metric /tildewideg∞ ijis cylindrical at any point q0∈γ∞which is suﬃciently close to y∞and then the metric space ¯V=V∪ {y∞}by adding the point y∞has nonnegative curvature in the Alexandrov sense. Consequently we have a three-dimensional non-ﬂat ta ngent cone Cy∞¯Vaty∞ which is a metric cone with aperture ≤20ε. On the other hand, note that by the canonical neighborhood as sumption, the canonical 2 ε-neck neighborhoods are strong. Thus at each point q∈Vneary∞, the limiting metric /tildewideg∞ ijactually exists on the whole parabolic neighborhood V/intersectiondisplay P/parenleftbigg q,0,1 3η−1(/tildewideR∞(q))−1 2,−1 10η−1(/tildewideR∞(q))−1/parenrightbigg , and is a smooth solution of the Ricci ﬂow there. Pick z∈Cy∞¯Vwith distance one from the vertex y∞and it is nonﬂat around z. By deﬁnition the ball B(z,1 2)⊂ Cy∞¯Vis the Gromov-Hausdorﬀ convergent limit of the scalings of a sequence of balls B∞(zℓ,σℓ)(⊂(V,/tildewideg∞ ij)) whereσℓ→0. Since the estimate (7.4.14) survives on ( V,/tildewideg∞ ij) for allA <+∞, and the tangent cone is three-dimensional and nonﬂat aroun dz, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 447 we see that this convergence is actually in the C∞ loctopology and over some ancient time interval. Since the limiting B∞(z,1 2)(⊂Cy∞¯V) is a piece of nonnegatively curved nonﬂat metric cone, we get a contradiction with Hamilton’s s trong maximum principle (Theorem 2.2.1) as before. So we have proved ρ0=∞. This proves (i). By the same proof of Assertion 1 in Step 2, we can further show t hat for any L, the rescaled solutions on the balls ˜B0(¯x,L) are deﬁned at least on the time interval [−1 16η−1C(L)−1,0] for all suﬃciently large k. This proves (ii). Step4. For any subsequence ( αk,βk) of (α,β) withrαk→0 and/tildewideδαkβk→0 ask→ ∞ , by Step 3, the κ-noncollapsing property and Hamilton’s compactness theorem, we can extract a C∞ locconvergent subsequence of ˜ gαkβk ijover some space-time open subsets containing the slice {t= 0}. We now want to show anysuch limit has bounded curvature at t= 0. We prove by contradiction. Suppose not, then there is a sequence of points zℓdivergent to inﬁnity in the limiting metric at time zero with curvature divergent to inﬁnity. Since the curvature at zℓis large (comparable to one),zℓhas a canonical neighborhood which is a 2 ε-cap or strong 2 ε-neck. Note that the boundary of 2 ε-cap lies in some 2 ε-neck. So we get a sequence of 2 ε-necks with radius going to zero. Note also that the limit has nonnegativ e sectional curvature. Without loss of generality, we may assume 2 ε<ε 0, whereε0is the positive constant in Proposition 6.1.1. Thus this arrives at a contradiction w ith Proposition 6.1.1. Step5. In this step, we will choose some subsequence ( αk,βk) of (α,β) so that we can extract a complete smooth limit of the rescaled soluti ons/tildewidegαkβk ijto the Ricci ﬂow with surgery on a time interval [ −a,0] for some a>0. Chooseαk,βk→ ∞ so thatrαk→0,/tildewideδαkβk→0, and Assertion 1, 2, 3 hold with α=αk,β=βkfor allA∈ {p/q|p,q= 1,2,...,k }, andB,C∈ {1,2,...,k }. By Step 3, we may assume the rescaled solutions /tildewidegαkβk ijconverge in the C∞ loctopology at the timet= 0. Since the curvature of the limit at t= 0 is bounded by Step 4, it follows from Assertion 1 in Step 2 and the choice of the sequence ( αk,βk) that the limiting (M∞,/tildewideg∞ ij(·,t)) is deﬁned at least on a backward time interval [ −a,0] for some positive constantaand is a smooth solution to the Ricci ﬂow there. Step6. We further want to extend the limit in Step 5 backwards in ti me to inﬁnity to get an ancient κ-solution. Let /tildewidegαkβk ijbe the convergent sequence obtained in the above Step 5. Denote by tmax= sup/braceleftig t′|we can take a smooth limit on ( −t′,0] (with bounded curvature at each time slice) from a subsequence of the rescaled solutions /tildewidegαkβk ij/bracerightig . We ﬁrst claim that there is a subsequence of the rescaled solu tions/tildewidegαkβk ijwhich con- verges in the C∞ loctopology to a smooth limit ( M∞,/tildewideg∞ ij(·,t)) on the maximal time interval ( −tmax,0]. Indeed, lettℓbe a sequence of positive numbers such that tℓ→tmaxand there exist smooth limits ( M∞,/tildewideg∞ ℓ(·,t)) deﬁned on ( −tℓ,0]. For each ℓ, the limit has nonnegative sectional curvature and has bounded curvature at each time s lice. Moreover by the gradient estimate in canonical neighborhood assumption wi th accuracy parameter 2 ε, the limit has bounded curvature on each subinterval [ −b,0]⊂(−tℓ,0]. Denote by /tildewideQ 448 H.-D. CAO AND X.-P. ZHU the scalar curvature upper bound of the limit at time zero ( /tildewideQis independent of ℓ). Then we can apply Li-Yau-Hamilton inequality (Corollary 2. 5.5) to get /tildewideR∞ ℓ(x,t)≤tℓ t+tℓ/tildewideQ, where/tildewideR∞ ℓ(x,t) are the scalar curvatures of the limits ( M∞,/tildewideg∞ ℓ(·,t)). Hence by the deﬁnition of convergence and the above curvature estimates , we can ﬁnd a subsequence of the rescaled solutions /tildewidegαkβk ijwhich converges in the C∞ loctopology to a smooth limit (M∞,/tildewideg∞ ij(·,t)) on the maximal time interval ( −tmax,0]. We need to show −tmax=−∞. Suppose −tmax>−∞, there are only the following two possibilities: either (1) The curvature of the limiting solution ( M∞,/tildewideg∞ ij(·,t)) becomes unbounded as tց −tmax; or (2) For each small constant θ>0 and each large integer k0>0, there is some k≥ k0such that the rescaled solution /tildewidegαkβk ijhas a surgery time Tk∈[−tmax−θ,0] and a surgery point xklying in a gluing cap at the times Tkso thatd2 Tk(xk,¯x) is uniformly bounded from above by a constant independent of θandk0. We next claim that the possibility (1) always occurs. Suppos e not; then the curvature of the limiting solution ( M∞,/tildewideg∞ ij(·,t)) is bounded on M∞×(−tmax,0] by some positive constant ˆC. In particular, for any A >0, there is a suﬃciently large integerk1>0 such that any rescaled solution /tildewidegαkβk ijwithk≥k1on the geodesic ball/tildewideB0(¯x,A) is deﬁned on the time interval [ −tmax+1 50η−1ˆC−1,0] and its scalar curvature is bounded by 2 ˆCthere. (Here, without loss of generality, we may assume that the upper bound ˆCis so large that −tmax+1 50η−1ˆC−1<0.) By Assertion 1 in Step 2, for klarge enough, the rescaled solution /tildewidegαkβk ijover/tildewideB0(¯x,A) can be deﬁned on the extended time interval [ −tmax−1 50η−1ˆC−1,0] and has the scalar curvature /tildewideR≤10ˆCon/tildewideB0(¯x,A)×[−tmax−1 50η−1ˆC−1,0]. So we can extract a smooth limit from the sequence to get the limiting solution which is deﬁne d on a larger time interval [−tmax−1 50η−1ˆC−1,0]. This contradicts the deﬁnition of the maximal time −tmax. It remains to exclude the possibility (1). By using Li-Yau-Hamilton inequality (Corollary 2.5.5) aga in, we have /tildewideR∞(x,t)≤tmax t+tmax/tildewideQ. So we only need to control the curvature near −tmax. Exactly as in Step 4 in the proof of Theorem 7.1.1, it follows from Li-Yau-Hamilton ine quality that (7.4.16) ˜d0(x,y)≤˜dt(x,y)≤˜d0(x,y) + 30tmax/radicalig /tildewideQ for anyx,y∈M∞andt∈(−tmax,0]. Since the inﬁmum of the scalar curvature is nondecreasing in time, we have some pointy∞∈M∞and some time −tmax<t∞<−tmax+1 50η−1such that/tildewideR∞(y∞,t∞)< 5/4. By (7.4.16), there is a constant /tildewideA0>0 such that ˜dt(¯x,y∞)≤/tildewideA0/2 for all t∈(−tmax,0]. Now we come back to the rescaled solution /tildewidegαkβk ij. Clearly, for arbitrarily given smallǫ′>0, whenklarge enough, there is a point ykin the underlying manifold of /tildewidegαkβk ijat time 0 satisfying the following properties (7.4.17) /tildewideR(yk,t∞)<3 2,/tildewidedt(¯x,yk)≤/tildewideA0 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 449 fort∈[−tmax+ǫ′,0]. By the deﬁnition of convergence, we know that for any ﬁxed A0≥2/tildewideA0, forklarge enough, the rescaled solution over /tildewideB0(¯x,A0) is deﬁned on the time interval [ t∞,0] and satisﬁes /tildewideR(x,t)≤2tmax t+tmax/tildewideQ on/tildewideB0(¯x,A0)×[t∞,0]. Then by Assertion 2 of Step 2, we have proved that there is a suﬃciently large integer ¯k0such that when k≥¯k0, the rescaled solutions /tildewidegαkβk ijatyk can be deﬁned on [ −tmax−1 50η−1,0], and satisfy /tildewideR(yk,t)≤15 fort∈[−tmax−1 50η−1,t∞]. We now prove a statement analogous to Assertion 4 (i) of Step 3 . Assertion 5. For the above rescaled solutions /tildewidegαkβk ijand¯k0, we have that for anyL >0, there is a positive constant ω(L) such that the rescaled solutions /tildewidegαkβk ij satisfy /tildewideR(x,t)≤ω(L) for all (x,t) with ˜dt(x,yk)≤Landt∈[−tmax−1 50η−1,t∞], and for all k≥¯k0. Proof of Assertion 5. We slightly modify the argument in the proof of Assertion 4 (i). Let M(ρ) = sup/braceleftig /tildewideR(x,t)|˜dt(x,yk)≤ρandt∈[−tmax−1 50η−1,t∞] in the rescaled solutions /tildewidegαkβk ij,k≥¯k0/bracerightig and ρ0= sup{ρ>0|M(ρ)<+∞}. Note that the estimate (7.4.14) implies that ρ0>0. We only need to show ρ0= +∞. We argue by contradiction. Suppose ρ0<+∞. Then, after passing to a subsequence, there is a sequence (˜ yk,tk) in the rescaled solutions /tildewidegαkβk ijwith tk∈[−tmax−1 50η−1,t∞] and ˜dtk(yk,˜yk)→ρ0<+∞such that/tildewideR(˜yk,tk)→+∞. De- note byγka minimizing geodesic segment from ykto ˜ykat the time tkand denote by /tildewideBtk(yk,ρ0) the open geodesic ball centered at ykof radiusρ0on the rescaled solution /tildewidegαkβk ij(·,tk). For any 0< ρ < ρ 0withρnearρ0, by applying Assertion 3 as before, we get that the rescaled solutions on the balls /tildewideBtk(yk,ρ) are deﬁned on the time interval [tk−1 16η−1M(ρ)−1,tk] for all large k. By Step 1 and Shi’s derivative estimate, we further know that the covariant derivatives and higher orde r derivatives of the curva- tures on/tildewideBtk(yk,ρ−(ρ0−ρ) 2)×[tk−1 32η−1M(ρ)−1,tk] are also uniformly bounded. Then by the uniform κ-noncollapsing property and Hamilton’s compactness theor em (The- orem 4.1.5), after passing to a subsequence, we can assume th at the marked sequence (˜Btk(yk,ρ0),/tildewidegαkβk ij(·,tk),yk) converges in the C∞ loctopology to a marked (noncomplete) 450 H.-D. CAO AND X.-P. ZHU manifold (B∞,/tildewideg∞ ij,y∞) and the geodesic segments γkconverge to a geodesic segment (missing an endpoint) γ∞⊂B∞emanating from y∞. Clearly, the limit also has nonnegative sectional curvatur e by the pinching as- sumption. Then by repeating the same argument as in the proof of Assertion 4 (i) in the rest, we derive a contradiction with Hamilton’s strong m aximum principle. This proves Assertion 5. We then apply the second estimate of (7.4.17) and Assertion 5 to conclude that for any large constant 0 <A< +∞, there is a positive constant C(A) such that for any smallǫ′>0, the rescaled solutions /tildewidegαkβk ijsatisfy (7.4.18) /tildewideR(x,t)≤C(A), for allx∈/tildewideB0(¯x,A) andt∈[−tmax+ǫ′,0], and for all suﬃciently large k. Then by applying Assertion 1 in Step 2, we conclude that the rescaled solutions/tildewidegαkβk ijon the geodesic balls /tildewideB0(¯x,A) are also deﬁned on the extended time interval [ −tmax+ǫ′− 1 8η−1C(A)−1,0] for all suﬃciently large k. Furthermore, by the gradient estimates as in Step 1, we have /tildewideR(x,t)≤10C(A), forx∈/tildewideB0(¯x,A) andt∈[−tmax+ǫ′−1 8η−1C(A)−1,0]. Sinceǫ′>0 is arbitrarily small and the positive constant C(A) is independent of ǫ′, we conclude that the rescaled solutions /tildewidegαkβk ijon/tildewideB0(¯x,A) are deﬁned on the extended time interval [ −tmax− 1 16η−1C(A)−1,0] and satisfy (7.4.19) /tildewideR(x,t)≤10C(A), forx∈/tildewideB0(¯x,A) andt∈[−tmax−1 16η−1C(A)−1,0], and for all suﬃciently large k. Now, by taking convergent subsequences from the rescaled so lutions/tildewidegαkβk ij, we see that the limit solution is deﬁned smoothly on a space-tim e open subset of M∞× (−∞,0] containing M∞×[−tmax,0]. By Step 4, we see that the limiting metric /tildewideg∞ ij(·,−tmax) at time −tmaxhas bounded curvature. Then by combining with the canonical neighborhood assumption of accuracy 2 ε, we conclude that the curvature of the limit is uniformly bounded on the time interval [ −tmax,0]. So we have excluded the possibility (1). Hence we have proved a subsequence of the rescaled solutions converges to an orientable ancient κ-solution. Finally by combining with the canonical neighborhood theor em (Theorem 6.4.6), we see that (¯ x,¯t) has a canonical neighborhood with parameter ε, which is a contra- diction. Therefore we have completed the proof of the propos ition. Summing up, we have proved that for any ε>0, (without loss of generality, we may assume ε≤¯ε0), there exist nonincreasing (continuous) positive functi ons/tildewideδ(t) and/tildewider(t), deﬁned on [0 ,+∞) with /tildewideδ(t)≤¯δ(t) = min/braceleftbigg1 2e2log(1 +t),δ0/bracerightbigg , such that for arbitrarily given (continuous) positive func tionδ(t) withδ(t)</tildewideδ(t) on [0,+∞), and arbitrarily given a compact orientable normalized th ree-manifold as THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 451 initial data, the Ricci ﬂow with surgery has a solution on [0 ,T) obtained by evolving the Ricci ﬂow and by performing δ-cutoﬀ surgeries at a sequence of times 0 < t1< t2<···< ti<···< T, withδ(ti)≤δ≤/tildewideδ(ti) at each time ti, so that the pinching assumption and the canonical neighborhood assumption (wit h accuracy ε) withr= /tildewider(t) are satisﬁed. (At this moment we still do not know whether th e surgery times ti are discrete.) Since theδ-cutoﬀ surgeries occur at the points lying deeply in the ε-horns, the minimum of the scalar curvature Rmin(t) of the solution to the Ricci ﬂow with surgery at each time-slice is achieved in the region unaﬀected by the surgeries. Thus we know from the evolution equation of the scalar curvature that (7.4.20)d dtRmin(t)≥2 3R2 min(t). In particular, the minimum of the scalar curvature Rmin(t) is nondecreasing in time. Also note that each δ-cutoﬀ surgery decreases volume. Then the upper derivative of the volume in time satisﬁes ¯/parenleftbiggd dt/parenrightbigg V(t)/defineslim sup △t→0V(t+△t)−V(t) △t ≤ −Rmin(0)V(t) which implies that V(t)≤V(0)e−Rmin(0)t. On the other hand, by Lemma 7.3.2 and the δ-cutoﬀ procedure given in the previous section, we know that at each time ti, eachδ-cutoﬀ surgery cuts down the volume at least at an amount of h3(ti) withh(ti) depending only on δ(ti) and/tildewider(ti). Thus the surgery times ticannot accumulate in any ﬁnite interval. When the solution becomes extinct at some ﬁnite time T, the solution at time near Tis entirely covered by canonical neighborhoods and then the initial manifold is di ﬀeomorphic to a connected sum of a ﬁnite copies of S2×S1andS3/Γ (the metric quotients of round three-sphere). So we have proved the following long-time existence result w hich was proposed by Perelman in [104]. Theorem 7.4.3 ( Long-time existence theorem ).For any ﬁxed constant ε >0, there exist nonincreasing (continuous )positive functions /tildewideδ(t)and/tildewider(t), deﬁned on [0,+∞), such that for an arbitrarily given (continuous )positive function δ(t)with δ(t)≤/tildewideδ(t)on[0,+∞), and arbitrarily given a compact orientable normalized thr ee- manifold as initial data, the Ricci ﬂow with surgery has a sol ution with the following properties: either (i) it is deﬁned on a ﬁnite interval [0,T)and obtained by evolving the Ricci ﬂow and by performing a ﬁnite number of cutoﬀ surgeries, with eac hδ-cutoﬀ at a timet∈(0,T)havingδ=δ(t), so that the solution becomes extinct at the ﬁnite timeT, and the initial manifold is diﬀeomorphic to a connected sum of a ﬁnite copies of S2×S1andS3/Γ (the metric quotients of round three-sphere ) ; or (ii) it is deﬁned on [0,+∞)and obtained by evolving the Ricci ﬂow and by per- forming at most countably many cutoﬀ surgeries, with each δ-cutoﬀ at a time t∈[0,+∞)havingδ=δ(t), so that the pinching assumption and the canoni- cal neighborhood assumption (with accuracy ε)withr=/tildewider(t)are satisﬁed, and there exist at most a ﬁnite number of surgeries on every ﬁnite time interval. 452 H.-D. CAO AND X.-P. ZHU In particular, if the initial manifold has positive scalar c urvature, say R≥a>0, then by (7.4.20), the solution becomes extinct at T≤3 2a. Hence we have the following topological description of compact three-manifolds with n onnegative scalar curvature which improves the well-known work of Schoen-Yau [109], [11 0]. Corollary 7.4.4 ( Perelman [104] ).LetMbe a compact orientable three- manifold with nonnegative scalar curvature. Then either Mis ﬂat or it is diﬀeomor- phic to a connected sum of a ﬁnite copies of S2×S1andS3/Γ (the metric quotients of the round three-sphere ). The famous Poincar´ e conjecture states that every compact three-manifold with trivial fundamental group is diﬀeomorphic to S3. Developing tools to attack the conjecture formed the basis for much of the works in three-di mensional topology over the last one hundred years. Now we use the Ricci ﬂow to dis cuss the Poincar´ e conjecture. LetMbe a compact three-manifold with trivial fundamental group . In particular, the three-manifold Mis orientable. Arbitrarily given a Riemannian metric on M, by scaling we may assume the metric is normalized. With this nor malized metric as initial data, we consider the solution to the Ricci ﬂow with s urgery. If one can show the solution becomes extinct in ﬁnite time, it will follow fr om Theorem 7.4.3 (i) that the three-manifold Mis diﬀeomorphic to the three-sphere S3. Such ﬁnite extinction time result was ﬁrst proposed by Perelman in [105], and recen tly, Colding-Minicozzi has published a proof to it in [42]. So the combination of Theorem 7.4.3 (i) and Colding-Minicozzi’s ﬁnite extinction result gives a compl ete proof of the Poincar´ e conjecture . We also remark that the above long-time existence result has been extended to compact four-manifolds with positive isotropic curvature by Chen and the second au- thor in [34]. As a consequence it gave a complete proof of the f ollowing classiﬁcation theorem of compact four-manifolds, with no essential incom pressible space-form and with a metric of positive isotropic curvature. The theorem w as ﬁrst proved by Hamil- ton in ([64]), though it was later found that the proof contai ns some gaps (see for example the comment of Perelman in Page 1, the second paragra ph, of [104]). Theorem 7.4.5. A compact four-manifold with no essential incompressible s pace- form and with a metric of positive isotropic curvature is diﬀ eomorphic to S4, orRP4, orS3×S1, orS3/tildewide×S1(theZ2quotient of S3×S1where Z2ﬂipsS3antipodally and rotates S1by1800),or a connected sum of them. 7.5. Curvature Estimates for Surgically Modiﬁed Solutions .In this sec- tion we will generalize the curvature estimates for smooth s olutions in Section 7.2 to that of solutions with cutoﬀ surgeries. We ﬁrst state and pro ve a version of Theorem 7.2.1. Theorem 7.5.1 ( Perelman [104] ).For anyε >0and1≤A < +∞, one can ﬁndκ=κ(A,ε)>0,K1=K1(A,ε)<+∞,K2=K2(A,ε)<+∞and¯r= ¯r(A,ε)>0such that for any t0<+∞there exists ¯δA=¯δA(t0)>0 (depending also onε),nonincreasing in t0, with the following property. Suppose we have a solution, constructed by Theorem 7.4.3with the nonincreasing (continuous) positive functions /tildewideδ(t)and/tildewider(t), to the Ricci ﬂow with δ-cutoﬀ surgeries on time interval [0,T]and with a compact orientable normalized three-manifold as initial data, where each δ-cutoﬀ at a timetsatisﬁesδ=δ(t)≤/tildewideδ(t)on[0,T]andδ=δ(t)≤¯δAon[t0 2,t0]; assume THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 453 that the solution is deﬁned on the whole parabolic neighborh oodP(x0,t0,r0,−r2 0)/defines {(x,t)|x∈Bt(x0,r0),t∈[t0−r2 0,t0]},2r2 0<t0, and satisﬁes |Rm| ≤r−2 0on P (x0,t0,r0,−r2 0), and Volt0(Bt0(x0,r0))≥A−1r3 0. Then (i) the solution is κ-noncollapsed on all scales less than r0in the ball Bt0(x0,Ar0); (ii) every point x∈Bt0(x0,Ar0)withR(x,t0)≥K1r−2 0has a canonical neighbor- hoodB, withBt0(x,σ)⊂B⊂Bt0(x,2σ)for some 0<σ <C 1(ε)R−1 2(x,t0), which is either a strong ε-neck or an ε-cap; (iii) ifr0≤¯r√t0thenR≤K2r−2 0inBt0(x0,Ar0). HereC1(ε)is the positive constant in the canonical neighborhood assu mption. Proof. Without loss of generality, we may assume 0 < ε≤¯ε0, where ¯ε0is the suﬃciently small (universal) positive constant in Lemma 7. 4.2. (i) This is analog of no local collapsing theorem II (Theorem 3.4.2). In comparison with the no local collapsing theorem II, this statement give sκ-noncollapsing property no matter how big the time is and it also allows the solution to be modiﬁed by surgery. Letη(≥10) be the universal constant in the deﬁnition of the canonic al neighbor- hood assumption. Recall that we had removed every component which has positive sectional curvature in our surgery procedure. By the same ar gument as in the ﬁrst part of the proof of Lemma 7.4.2, the canonical neighborhood assumption of the so- lution implies the κ-noncollapsing on the scales less than1 2η/tildewider(t0) for some positive constantκdepending only on C1(ε) andC2(ε) (in the deﬁnition of the canonical neighborhood assumption). So we may assume1 2η/tildewider(t0)≤r0≤/radicalig t0 2, and study the scalesρ,1 2η/tildewider(t0)≤ρ≤r0. Letx∈Bt0(x0,Ar0) and assume that the solution satisﬁes |Rm| ≤ρ−2 for those points in P(x,t0,ρ,−ρ2)/defines{(y,t)|y∈Bt(x,ρ),t∈[t0−ρ2,t0]}for which the solution is deﬁned. We want to bound the ratio Vol t0(Bt0(x,ρ))/ρ3from below. Recall that a space-time curve is called admissible if it sta ys in the region unaf- fected by surgery, and a space-time curve on the boundary of t he set of admissible curves is called a barely admissible curve. Consider any bar ely admissible curve γ, parametrized by t∈[tγ,t0],t0−r2 0≤tγ≤t0, withγ(t0) =x. The same proof for the assertion (7.4.2) (in the proof of Lemma 7.4.2) shows that fo r arbitrarily large L>0 (to be determined later), one can ﬁnd a suﬃciently small ¯δ(L,t0,/tildewider(t0),/tildewider(t0 2),ε)>0 such that when each δ-cutoﬀ in [t0 2,t0] satisﬁesδ≤¯δ(L,t0,/tildewider(t0),/tildewider(t0 2),ε), there holds (7.5.1)/integraldisplayt0 tγ√t0−t(R+(γ(t),t) +|˙γ(t)|2)dt≥Lr0. From now on, we assume that each δ-cutoﬀ of the solution in the time interval [t0 2,t0] satisﬁesδ≤¯δ(L,t0,/tildewider(t0),/tildewider(t0 2),ε). 454 H.-D. CAO AND X.-P. ZHU Let us scale the solution, still denoted by gij(·,t), to make r0= 1 and the time ast0= 1. By the maximum principle, it is easy to see that the (resca led) scalar curvature satisﬁes R≥ −3 2t on (0,1]. Let us consider the time interval [1 2,1] and deﬁne a function of the form h(y,t) =φ(dt(x0,y)−A(2t−1))(¯L(y,τ) + 2√τ) whereτ= 1−t,φis the function of one variable chosen in the proof of Theorem 3.4.2 which is equal to one on ( −∞,1 20), rapidly increasing to inﬁnity on (1 20,1 10), and satisﬁes 2(φ′)2 φ−φ′′≥(2A+ 300)φ′−C(A)φfor some constant C(A)<+∞, and ¯L is the function deﬁned by ¯L(q,τ) = inf/braceleftig 2√τ/integraldisplayτ 0√s(R+|˙γ|2)ds|(γ(s),s),s∈[0,τ] is a space-time curve with γ(0) =xandγ(τ) =q/bracerightig . Note that ¯L(y,τ)≥2√τ/integraldisplayτ 0√sRds (7.5.2) ≥ −4τ2 >−2√τ sinceR≥ −3 and 0<τ≤1 2. This says his positive for t∈[1 2,1]. Also note that (7.5.3)∂ ∂τ¯L+△¯L≤6 as long as ¯Lis achieved by admissible curves. Then as long as the shortes tL-geodesics from (x0,0) to (y,τ) are admissible, there holds at yandt= 1−τ, /parenleftbigg∂ ∂t− △/parenrightbigg h≥/parenleftbigg φ′/bracketleftbigg/parenleftbigg∂ ∂t− △/parenrightbigg dt−2A/bracketrightbigg −φ′′/parenrightbigg ·(¯L+ 2√τ) −/parenleftbigg 6 +1√τ/parenrightbigg φ−2/a\}b∇acketle{t∇φ,∇¯L/a\}b∇acket∇i}ht. Firstly, we may assume the constant Lin (7.5.1) is not less than 2 exp(C(A) + 100). We claim that Lemma 3.4.1(i) is applicable for d=dt(·,x0) atyandt= 1−τ(withτ∈[0,1 2]) whenever ¯L(y,τ) is achieved by admissible curves and satisﬁes the estimate ¯L(y,τ)≤3√τexp(C(A) + 100). Indeed, since the solution is deﬁned on the whole neighborho odP(x0,t0,r0,−r2 0) with r0= 1 andt0= 1, the point x0at the time t= 1−τlies on the region unaﬀected by surgery. Note that R≥ −3 fort∈[1 2,1]. When ¯L(y,τ) is achieved by admissible curves and satisﬁes ¯L(y,τ)≤3√τexp(C(A) + 100), the estimate (7.5.1) implies that THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 455 the pointyat the time t= 1−τdoes not lie in the collars of the gluing caps. Thus any minimal geodesic (with respect to the metric gij(·,t) witht= 1−τ) connecting x0andyalso lies in the region unaﬀected by surgery; otherwise the g eodesic is not minimal. Then from the proof of Lemma 3.4.1(i), we see that it is applicable. Assuming the minimum of hat a time, say t= 1−τ, is achieved at a point, sayy, and assuming ¯L(y,τ) is achieved by admissible curves and satisﬁes ¯L(y,τ)≤ 3√τexp(C(A) + 100), we have (¯L+ 2√τ)∇φ=−φ∇¯L, and then by the computations and estimates in the proof of The orem 3.4.2, /parenleftbigg∂ ∂t− △/parenrightbigg h ≥/parenleftbigg φ′/bracketleftbigg/parenleftbigg∂ ∂t− △/parenrightbigg dt−2A/bracketrightbigg −φ′′+ 2(φ′)2 φ/parenrightbigg ·(¯L+ 2√τ)−/parenleftbigg 6 +1√τ/parenrightbigg φ ≥ −C(A)h−/parenleftbigg 6 +1√τ/parenrightbiggh (2√τ−4τ2), atyandt= 1−τ. Here we used (7.5.2) and Lemma 3.4.1(i). As before, denoting by hmin(τ) = min zh(z,1−τ), we obtain d dτ/parenleftbigg log/parenleftbigghmin(τ)√τ/parenrightbigg/parenrightbigg ≤C(A) +6√τ+ 1 2τ−4τ2√τ−1 2τ(7.5.4) ≤C(A) +50√τ, as long as the associated shortest L-geodesics are admissible with ¯L≤3√τexp(C(A)+ 100). On the other hand, by deﬁnition, we have (7.5.5) lim τ→0+hmin(τ)√τ≤φ(d1(x0,x)−A)·2 = 2. The combination of (7.5.4) and (7.5.5) gives the following a ssertion: Letτ∈[0,1 2]. If for each s∈[0,τ],inf{¯L(y,s)|dt(x0,y)≤A(2t−1) + 1 10withs= 1−t}is achieved by admissible curves, then we have inf/braceleftbigg ¯L(y,τ)|dt(x0,y)≤A(2t−1) +1 10withτ= 1−t/bracerightbigg (7.5.6) ≤2√τexp(C(A) + 100). Note again that R≥ −3 fort∈[1 2,1]. By combining with (7.5.1), we know that any barely admissible curve γ, parametrized by s∈[0,τ], 0≤τ≤1 2, withγ(0) =x, satisﬁes /integraldisplayτ 0√s(R+|˙γ|2)ds≥7 4exp(C(A) + 100), by assuming L≥2 exp(C(A) + 100). 456 H.-D. CAO AND X.-P. ZHU Since |Rm| ≤ρ−2onP(x,t0,ρ,−ρ2) withρ≥1 2η/tildewider(t0) (andt0= 1) and¯δ(L,t0,/tildewider(t0),/tildewider(t0 2),ε)>0 is suﬃciently small, the parabolic neighborhood P(x,1,ρ,−ρ2) around the point ( x,1) is contained in the region unaﬀected by the surgery. Thus as τ= 1−tis suﬃciently close to zero,1 2√τinf¯Lcan be bounded from above by a small positive constant and then the inﬁmum inf {¯L(y,τ)|dt(x0,y)≤ A(2t−1) +1 10withτ= 1−t}is achieved by admissible curves. Hence we conclude that for each τ∈[0,1 2], any minimizing curve γτof inf{¯L(y,τ)|dt(x0,y)≤A(2t−1) +1 10withτ= 1−t}is admissible and satisﬁes /integraldisplayτ 0√s(R+|˙γτ|2)ds≤exp(C(A) + 100). Now we come back to the unrescaled solution. It then follows t hat the Li-Yau- Perelman distance lfrom (x,t0) satisﬁes the following estimate (7.5.7) min/braceleftbigg l/parenleftbigg y,t0−1 2r2 0/parenrightbigg/vextendsingle/vextendsingle/vextendsingley∈Bt0−1 2r2 0/parenleftbigg x0,1 10r0/parenrightbigg/bracerightbigg ≤exp(C(A) + 100), by noting the (parabolic) scaling invariance of the Li-Yau- Perelman distance. By the assumption that |Rm| ≤r−2 0onP(x0,t0,r0,−r2 0), exactly as before, for anyq∈Bt0−r2 0(x0,r0), we can choose a path γparametrized by τ∈[0,r2 0] with γ(0) =x,γ(r2 0) =q, andγ(1 2r2 0) =y∈Bt0−1 2r2 0(x0,1 10r0), whereγ|[0,1 2r2 0]achieves the minimum min {l(y,t0−1 2r2 0)|y∈Bt0−1 2r2 0(x0,1 10r0)}andγ|[1 2r2 0,r2 0]is a suitable curve satisfyingγ|[1 2r2 0,r2 0](τ)∈Bt0−τ(x0,r0), for each τ∈[1 2r2 0,r2 0], so that the L-length ofγis uniformly bounded from above by a positive constant (depe nding only on A) multiplying r0. This implies that the Li-Yau-Perelman distance from ( x,t0) to the ballBt0−r2 0(x0,r0) is uniformly bounded by a positive constant L(A) (depending only onA). Now we can choose the constant Lin (7.5.1) by L= max {2L(A),2 exp(C(A) + 100) }. Thus every shortest L-geodesic from ( x,t0) to the ball Bt0−r2 0(x0,r0) is necessarily admissible. By combining with the assumption that Vol t0(Bt0(x0,r0))≥A−1r3 0, we conclude that Perelman’s reduced volume of the ball Bt0−r2 0(x0,r0) satisﬁes the estimate /tildewideVr2 0(Bt0−r2 0(x0,r0)) =/integraldisplay Bt0−r2 0(x0,r0)(4πr2 0)−3 2exp(−l(q,r2 0))dVt0−r2 0(q) (7.5.8) ≥c(A) for some positive constant c(A) depending only on A. We can now argue as in the last part of the proof of Lemma 7.4.2 t o get a lower bound estimate for the volume of the ball Bt0(x,ρ). The union of all shortest L- geodesics from ( x,t0) to the ball Bt0−r2 0(x0,r0), deﬁned by CBt0−r2 0(x0,r0) ={(y,t)|(y,t) lies in a shortest L-geodesic from (x,t0) to a point in Bt0−r2 0(x0,r0)}, forms a cone-like subset in space-time with vertex ( x,t0). Denote by B(t) the inter- section of the cone-like subset CBt0−r2 0(x0,r0) with the time-slice at t. Perelman’s THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 457 reduced volume of the subset B(t) is given by /tildewideVt0−t(B(t)) =/integraldisplay B(t)(4π(t0−t))−3 2exp(−l(q,t0−t))dVt(q). Since the cone-like subset CBt0−r2 0(x0,r0) lies entirely in the region unaﬀected by surgery, we can apply Perelman’s Jacobian comparison Theor em 3.2.7 and the esti- mate (7.5.8) to conclude that /tildewideVt0−t(B(t))≥/tildewideVr2 0(Bt0−r2 0(x0,r0)) (7.5.9) ≥c(A) for allt∈[t0−r2 0,t0]. As before, denoting by ξ=ρ−1Volt0(Bt0(x,ρ))1 3, we only need to get a positive lower bound for ξ. Of course we may assume ξ<1. Consider/tildewideB(t0−ξρ2), the subset at the time-slice {t=t0−ξρ2}where every point can be connected to ( x,t0) by an admissible shortest L-geodesic. Perelman’s reduced volume of /tildewideB(t0−ξρ2) is given by /tildewideVξρ2(/tildewideB(t0−ξρ2)) (7.5.10) =/integraldisplay/CTB(t0−ξρ2)(4πξρ2)−3 2exp(−l(q,ξρ2))dVt0−ξρ2(q) =/integraldisplay/CTB(t0−ξρ2)∩Lexp {|υ|≤1 4ξ−1 2}(ξρ2)(4πξρ2)−3 2exp(−l(q,ξρ2))dVt0−ξρ2(q) +/integraldisplay/CTB(t0−ξρ2)\Lexp {|υ|≤1 4ξ−1 2}(ξρ2)(4πξρ2)−3 2exp(−l(q,ξρ2))dVt0−ξρ2(q). Note that the whole region P(x,t0,ρ,−ρ2) is unaﬀected by surgery because ρ≥ 1 2η/tildewider(t0) and ¯δ(L,t0,/tildewider(t0),/tildewider(t0 2),ε)>0 is suﬃciently small. Then exactly as before, there is a universal positive constant ξ0such that when 0 <ξ≤ξ0, there holds Lexp{|υ|≤1 4ξ−1 2}(ξρ2)⊂Bt0(x,ρ) and the ﬁrst term on RHS of (7.5.10) can be estimated by /integraldisplay/CTB(t0−ξρ2)∩Lexp {|υ|≤1 4ξ−1 2}(ξρ2)(4πξρ2)−3 2exp(−l(q,ξρ2))dVt0−ξρ2(q) (7.5.11) ≤eCξ(4π)−3 2ξ3 2 for some universal constant C; while the second term on RHS of (7.5.10) can be estimated by /integraldisplay/CTB(t0−ξρ2)\Lexp {|υ|≤1 4ξ−1 2}(ξρ2)(4πξρ2)−3 2exp(−l(q,ξρ2))dVt0−ξρ2(q) (7.5.12) ≤(4π)−3 2/integraldisplay {|υ|>1 4ξ−1 2}exp(−|υ|2)dυ. SinceB(t0−ξρ2)⊂/tildewideB(t0−ξρ2), the combination of (7.5.9)-(7.5.12) bounds ξfrom below by a positive constant depending only on A. This proves the statement (i). 458 H.-D. CAO AND X.-P. ZHU (ii) This is analogous to the claim in the proof of Theorem 7.2 .1. We argue by contradiction. Suppose that for some A<+∞and a sequence Kα 1→ ∞, there exists a sequence tα 0such that for any sequences ¯δαβ>0 with ¯δαβ→0 for ﬁxed α, we have sequences of solutions gαβ ijto the Ricci ﬂow with surgery and sequences of points xαβ 0, of radiirαβ 0, which satisfy the assumptions but violate the statement (i i) at some xαβ∈Btα 0(xαβ 0,Arαβ 0) withR(xαβ,tα 0)≥Kα 1(rαβ 0)−2. Slightly abusing notation, we will often drop the indices α,βin the following argument. Exactly as in the proof of Theorem 7.2.1, we need to adjust the point (x,t0). More precisely, we claim that there exists a point (¯ x,¯t)∈B¯t(x0,2Ar0)×[t0−r2 0 2,t0] with ¯Q/defines R(¯x,¯t)≥K1r−2 0such that the point (¯ x,¯t) does not satisfy the canonical neighborhood statement, but each point ( y,t)∈¯PwithR(y,t)≥4¯Qdoes, where ¯Pis the set of all (x′,t′) satisfying ¯t−1 4K1¯Q−1≤t′≤¯t,dt′(x0,x′)≤d¯t(x0,¯x) +K1 2 1¯Q−1 2. Indeed as before, the point (¯ x,¯t) is chosen by an induction argument. We ﬁrst choose ( x1,t1) = (x,t0) which satisﬁes dt1(x0,x1)≤Ar0andR(x1,t1)≥K1r−2 0, but does not satisfy the canonical neighborhood statement. Now if ( xk,tk) is already chosen and is not the desired (¯x,¯t), then some point ( xk+1,tk+1) satisﬁestk−1 4K1R(xk,tk)−1≤tk+1≤tk, dtk+1(x0,xk+1)≤dtk(x0,xk) +K1 2 1R(xk,tk)−1 2, andR(xk+1,tk+1)≥4R(xk,tk), but (xk+1,tk+1) does not satisfy the canonical neighborhood statement. Th en we have R(xk+1,tk+1)≥4kR(x1,t1)≥4kK1r−2 0, dtk+1(x0,xk+1)≤dt1(x0,x1) +K1 2 1k/summationdisplay i=1R(xi,ti)−1 2≤Ar0+ 2r0, and t0≥tk+1≥t0−1 4K1k/summationdisplay i=1R(xi,ti)−1≥t0−1 2r2 0. So the sequence must be ﬁnite and its last element is the desir ed (¯x,¯t). Rescale the solutions along (¯ x,¯t) with factor R(¯x,¯t)(≥K1r−2 0) and shift the times ¯tto zero. We will adapt both the proof of Proposition 7.4.1 and that of Theorem 7.2.1 to show that a sequence of the rescaled solutions /tildewidegαβ ijconverges to an ancient κ-solution, which will give the desired contradiction. Sinc e we only need to consider the scale of the curvature less than /tildewider(¯t)−2, the present situation is much easier than that of Proposition 7.4.1. Firstly as before, we need to get a local curvature estimate. For each adjusted (¯ x,¯t), let [t′,¯t] be the maximal subinterval of [ ¯t−1 20η−1¯Q−1,¯t] so that for each suﬃciently large αand then suﬃciently large β, the canonical neighborhood statement holds for any ( y,t) inP(¯x,¯t,1 10K1 2 1¯Q−1 2,t′−¯t) ={(x,t)|x∈ Bt(¯x,1 10K1 2 1¯Q−1 2),t∈[t′,¯t]}withR(y,t)≥4¯Q, whereηis the universal positive constant in the deﬁnition of canonical neighborhood assump tion. We want to show (7.5.13) t′=¯t−1 20η−1¯Q−1. Consider the scalar curvature Rat the point ¯ xover the time interval [ t′,¯t]. If there is a time ˜t∈[t′,¯t] satisfying R(¯x,˜t)≥4¯Q, we let ˜tbe the ﬁrst of such time from ¯t. Since the chosen point (¯ x,¯t) does not satisfy the canonical neighborhood statement, THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 459 we knowR(¯x,¯t)≤/tildewider(¯t)−2. Recall from our designed surgery procedure that if there is a cutoﬀ surgery at a point xat a timet, the scalar curvature at ( x,t) is at least (¯δαβ)−2/tildewider(t)−2. Then for each ﬁxed α, forβlarge enough, the solution gαβ ij(·,t) around the point ¯xover the time interval [ ˜t−1 20η−1¯Q−1,˜t] is well deﬁned and satisﬁes the following curvature estimate R(¯x,t)≤8¯Q, fort∈[˜t−1 20η−1¯Q−1,¯t] (ort∈[t′,¯t] if there is no such time ˜t). By the assumption thatt0>2r2 0, we have ¯tR(¯x,¯t)≥t0 2R(¯x,¯t) ≥r2 0(K1r−2 0) =K1→+∞. Thus by using the pinching assumption and the gradient estim ates in the canonical neighborhood assumption, we further have |Rm(x,t)| ≤30¯Q, for allx∈Bt(¯x,1 10η−1¯Q−1 2) andt∈[˜t−1 20η−1¯Q−1,¯t] (ort∈[t′,¯t]) and all suﬃciently largeαandβ. Observe that Lemma 3.4.1 (ii) is applicable for dt(x0,¯x) witht∈ [˜t−1 20η−1¯Q−1,¯t] (ort∈[t′,¯t]) since any minimal geodesic, with respect to the metric gij(·,t), connecting x0and ¯xlies in the region unaﬀected by surgery; otherwise the geodesic is not minimal. After having obtained the above cur vature estimate, we can argue as deriving (7.2.2) and (7.2.3) in the proof of Theorem 7.2.1 to conclude that any point ( x,t), with ˜t−1 20η−1¯Q−1≤t≤¯t(ort∈[t′,¯t]) anddt(x,¯x)≤1 10K1 2 1¯Q−1 2, satisﬁes dt(x,x0)≤d¯t(¯x,x0) +1 2K1 2 1¯Q−1 2, for all suﬃciently large αandβ. Then by combining with the choice of the points (¯x,¯t), we prove t′=¯t−1 20η−1¯Q−1(i.e., the canonical neighborhood statement holds for any point ( y,t) in the parabolic neighborhood P(¯x,¯t,1 10K1 2 1¯Q−1 2,−1 20η−1¯Q−1) with R(y,t)≥4¯Q) for all suﬃciently large αand then suﬃciently large β. Now it follows from the gradient estimates in the canonical n eighborhood assump- tion that the scalar curvatures of the rescaled solutions /tildewidegαβ ijsatisfy /tildewideR(x,t)≤40 for those ( x,t)∈P(¯x,0,1 10η−1,−1 20η−1)/defines{(x′,t′)|x′∈/tildewideBt′(¯x,1 10η−1),t′∈ [−1 20η−1,0]}, for which the rescaled solution is deﬁned. (Here /tildewideBt′denotes the geodesic ball in the rescaled solution at time t′). Note again that R(¯x,¯t)≤/tildewider(¯t)−2and recall from our designed surgery procedure that if there is a cutoﬀ s urgery at a point xat a timet, the scalar curvature at ( x,t) is at least ( ¯δαβ)−2/tildewider(t)−2. Then for each ﬁxed suﬃciently large α, forβlarge enough, the rescaled solution /tildewidegαβ ijis deﬁned on the whole parabolic neighborhood P(¯x,0,1 10η−1,−1 20η−1). More generally, for arbitrarily ﬁxed 0</tildewideK < +∞, there is a positive integer α0so that for each α≥α0we can 460 H.-D. CAO AND X.-P. ZHU ﬁndβ0>0 (depending on α) such that if β≥β0and (y,0) is a point on the rescaled solution/tildewidegαβ ijwith/tildewideR(y,0)≤/tildewideKand/tildewided0(y,¯x)≤/tildewideK, we have estimate (7.5.14) /tildewideR(x,t)≤40/tildewideK for (x, t)∈P(y,0,1 10η−1/tildewideK−1 2,−1 20η−1/tildewideK−1)/defines{(x′,t′)|x′∈/tildewideBt′(y, 1 10η−1/tildewideK−1 2),t′∈[−1 20η−1/tildewideK−1,0]}. In particular, the rescaled solution is deﬁned on the whole parabolic neighborhood P(y,0,1 10η−1/tildewideK−1 2,−1 20η−1/tildewideK−1). Next, we want to show the curvature of the rescaled solutions at the new times zero (after shifting) stay uniformly bounded at bounded dis tances from ¯ xfor some subsequences of αandβ. Letαm,βm→+∞be chosen so that the estimate (7.5.14) holds with/tildewideK=m. For allρ≥0, set M(ρ) = sup {/tildewideR(x,0)|m≥1,d0(x,¯x)≤ρin the rescaled solutions /tildewidegαmβm ij} and ρ0= sup{ρ≥0|M(ρ)<+∞}. Clearly the estimate (7.5.14) yields ρ0>0. As we consider the unshifted time ¯t, by combining with the assumption that t0>2r2 0, we have ¯tR(¯x,¯t)≥t0 2R(¯x,¯t) (7.5.15) ≥r2 0(K1r−2 0) =K1→+∞. It then follows from the pinching assumption that we only nee d to showρ0= +∞. As before, we argue by contradiction. Suppose we have a seque nce of points ymin the rescaled solutions /tildewidegαmβm ij with/tildewided0(¯x,ym)→ρ0<+∞and/tildewideR(ym,0)→+∞. De- note byγma minimizing geodesic segment from ¯ xtoymand denote by /tildewideB0(¯x,ρ0) the open geodesic balls centered at ¯ xof radiusρ0of the rescaled solutions. By applying the assertion in statement (i), we have uniform κ-noncollapsing at the points (¯ x,¯t). By combining with the local curvature estimate (7.5.14) and Hamilton’s compactness theorem, we can assume that, after passing to a subsequence, the marked sequence (/tildewideB0(¯x,ρ0),/tildewidegαmβm ij,¯x) converges in the C∞ loctopology to a marked (noncomplete) man- ifold (B∞,/tildewideg∞ ij,x∞) and the geodesic segments γmconverge to a geodesic segment (missing an endpoint) γ∞⊂B∞emanating from x∞. Moreover, by the pinching assumption and the estimate (7.5.15), the limit has nonnega tive sectional curvature. Then exactly as before, we consider the tubular neighborhoo d alongγ∞ V=/uniondisplay q0∈γ∞B∞(q0,4π(/tildewideR∞(q0))−1 2) and the completion ¯B∞of (B∞,/tildewideg∞ ij) withy∞∈¯B∞the limit point of γ∞. As before, by the choice of the points (¯ x,¯t), we know that the limiting metric /tildewideg∞ ijis cylindrical at any point q0∈γ∞which is suﬃciently close to y∞. Then by the same reason as before the metric space ¯V=V∪ {y∞}has nonnegative curvature in Alexandrov sense, and we have a three-dimensional nonﬂat ta ngent cone Cy∞¯Vat y∞. Pickz∈Cy∞¯Vwith distance one from the vertex and it is nonﬂat around z. By THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 461 deﬁnitionB(z,1 2)(⊂Cy∞¯V) is the Gromov-Hausdoﬀ convergent limit of the scalings of a sequence of balls B∞(zk,σk)(⊂(V,/tildewideg∞ ij)) withσk→0. Since the estimate (7.5.14) survives on ( V,/tildewideg∞ ij) for all/tildewideK < +∞, we know that this convergence is actually in theC∞ loctopology and over some time interval. Since the limit B(z,1 2)(⊂Cy∞¯V) is a piece of a nonnegatively curved nonﬂat metric cone, we get a contradiction with Hamilton’s strong maximum Principle (Theorem 2.2.1) as bef ore. Hence we have proved that a subsequence of the rescaled solution /tildewidegαmβm ij has uniformly bounded curvatures at bounded distance from ¯ xat the new times zero. Further, by the uniform κ-noncollapsing at the points (¯ x,¯t) and the estimate (7.5.14) again, we can take a C∞ loclimit (M∞,/tildewideg∞ ij,x∞), deﬁned on a space-time subset which contains the time slice {t= 0}and is relatively open in M∞×(−∞,0], for the subsequence of the rescaled solutions. The limit is a smo oth solution to the Ricci ﬂow, and is complete at t= 0, as well as has nonnegative sectional curvature by the pinching assumption and the estimate (7.5.15). Thus by repe ating the same argument as in the Step 4 of the proof Proposition 7.4.1, we conclude th at the curvature of the limit att= 0 is bounded. Finally we try to extend the limit backwards in time to get an a ncientκ-solution. Since the curvature of the limit is bounded at t= 0, it follows from the estimate (7.5.14) that the limit is a smooth solution to the Ricci ﬂow d eﬁned at least on a backward time interval [ −a,0] for some positive constant a. Let (t∞,0] be the maximal time interval over which we can extract a smooth limiting sol ution. It suﬃces to show t∞=−∞. Ift∞>−∞, there are only two possibilities: either there exist surge ries in ﬁnite distance around the time t∞or the curvature of the limiting solution becomes unbounded as tցt∞. Letc>0 be a positive constant much smaller than1 100η−1. Note again that the inﬁmum of the scalar curvature is nondecreasing in time. The n we can ﬁnd some point y∞∈M∞and some time t=t∞+θwith 0<θ<c 3such that/tildewideR∞(y∞,t∞+θ)≤2. Consider the (unrescaled) scalar curvature Rofgαmβm ij(·,t) at the point ¯ xover the time interval [ ¯t+ (t∞+θ 2)¯Q−1,¯t]. Since the scalar curvature R∞of the limit on M∞×[t∞+θ 3,0] is uniformly bounded by some positive constant C, we have the curvature estimate R(¯x,t)≤2C¯Q for allt∈[¯t+ (t∞+θ 2)¯Q−1,¯t] and all suﬃciently large m. For each ﬁxed mandαm, we may require the chosen βmto satisfy (¯δαmβm)−2/parenleftbigg /tildewider/parenleftbiggtαm 0 2/parenrightbigg/parenrightbigg−2 ≥m/tildewider(tαm 0)−2≥m¯Q. Whenmis large enough, we observe again that Lemma 3.4.1 (ii) is app licable for dt(x0,¯x) witht∈[¯t+ (t∞+θ 2)¯Q−1,¯t]. Then by repeating the argument as in the derivation of (7.2.1), (7.2.2) and (7.2.3), we deduce that f or all suﬃciently large m, the canonical neighborhood statement holds for any ( y,t) in the parabolic neighborhood P(¯x,¯t,1 10K1 2 1¯Q−1 2,(t∞+θ 2)¯Q−1). Let (ym,¯t+ (t∞+θm)¯Q−1) be a sequence of associated points and times in the (unrescaled) solutions gαmβm ij(·,t) so that after rescaling, the sequence converges to (y∞,t∞+θ) in the limit. Clearlyθ 2≤θm≤2θfor all suﬃciently large m. Then by the argument as in the derivation of (7.5.13), we know that fo r all suﬃciently large 462 H.-D. CAO AND X.-P. ZHU m, the solution gαmβm ij(·,t) atymis deﬁned on the whole time interval [ ¯t+(t∞+θm− 1 20η−1)¯Q−1,¯t+ (t∞+θm)¯Q−1] and satisﬁes the curvature estimate R(ym,t)≤8¯Q there; moreover the canonical neighborhood statement hold s for any ( y,t) with R(y,t)≥4¯Qin the parabolic neighborhood P(ym,¯t,1 10K1 2 1¯Q−1 2,(t∞−c 3)¯Q−1). We now consider the rescaled sequence /tildewidegαmβm ij(·,t) with the marked points re- placed byymand the times replaced by sm∈[¯t+ (t∞−c 4)¯Q−1,¯t+ (t∞+c 4)¯Q−1]. As before the Li-Yau-Hamilton inequality implies the resca ling limit around ( ym,sm) agrees with the original one. Then the arguments in previous paragraphs imply the limit is well-deﬁned and smooth on a space-time open neighbo rhood of the maximal time slice {t=t∞}. Particularly this excludes the possibility of existing su rgeries in ﬁnite distance around the time t∞. Moreover, the limit at t=t∞also has bounded curvature. By using the gradient estimates in the canonical neighborhood assump- tion on the parabolic neighborhood P(ym,¯t,1 10K1 2 1¯Q−1 2,(t∞−c 3)¯Q−1), we see that the second possibility is also impossible. Hence we have pro ved a subsequence of the rescaled solutions converges to an ancient κ-solution. Therefore we have proved the canonical neighborhood statem ent (ii). (iii) This is analogous to Theorem 7.2.1. We also argue by con tradiction. Suppose for someA<+∞and sequences of positive numbers Kα 2→+∞, ¯rα→0 there exists a sequence of times tα 0such that for any sequences ¯δαβ>0 with ¯δαβ→0 for ﬁxedα, we have sequences of solutions gαβ ijto the Ricci ﬂow with surgery and sequences of points xαβ 0, of positive constants rαβ 0withrαβ 0≤¯rα/radicalbigtα 0which satisfy the assumptions, but for allα,βthere hold (7.5.16) R(xαβ,tα 0)>Kα 2(rαβ 0)−2,for somexαβ∈Btα 0(xαβ 0,Arαβ 0). We may assume that ¯δαβ≤¯δ4A(tα 0) for allα,β, where ¯δ4A(tα 0) is chosen so that the statements (i) and (ii) hold on Btα 0(xαβ 0,4Arαβ 0). Let ˆgαβ ijbe the rescaled solutions ofgαβ ijaround the origins xαβ 0with factor ( rαβ 0)−2and shift the times tα 0to zero. Then by applying the statement (ii), we know that the regions, whe re the scalar curvature of the rescaled solutions ˆ gαβ ijis at leastK1(=K1(4A)), are canonical neighborhood regions. Note that canonical ε-neck neighborhoods are strong. Also note that the pinching assumption and the assertion tα 0(rαβ 0)−2≥(¯rα)−2→+∞,asα→+∞, imply that any subsequent limit of the rescaled solutions ˆ gαβ ijmust have nonnegative sectional curvature. Thus by the above argument in the proof of the statement (ii) (or the argument in Step 2 of the proof of Theorem 7.1.1), we co nclude that there exist subsequences α=αm,β=βmsuch that the curvatures of the rescaled solutions ˆgαmβm ij stay uniformly bounded at distances from the origins xαmβm 0 not exceeding 2 A. This contradicts (7.5.16) for msuﬃciently large. This proves the statements (iii). Clearly for ﬁxed A, after deﬁning the ¯δA(t0) for eacht0, one can adjust the ¯δA(t0) so that it is nonincreasing in t0. Therefore we have completed the proof of the theorem. From now on we redeﬁne the function /tildewideδ(t) so that it is also less than ¯δ2(t+1)(2t) and then the above theorem always holds for A∈[1,2(t0+ 1)]. Particularly, we still THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 463 have /tildewideδ(t)≤¯δ(t) = min {1 2e2log(1 +t),δ0}, which tends to zero as t→+∞. We may also require that /tildewider(t) tends to zero as t→+∞. The next result is a version of Theorem 7.2.2 for solutions wi th surgery. Theorem 7.5.2 ( Perelman [104] ).For anyε >0andw >0, there exist τ= τ(w,ε)>0,K=K(w,ε)<+∞,¯r= ¯r(w,ε)>0,θ=θ(w,ε)>0andT= T(w)<+∞with the following property. Suppose we have a solution, con structed by Theorem 7.4.3with the nonincreasing (continuous )positive functions /tildewideδ(t)and/tildewider(t), to the Ricci ﬂow with surgery on the time interval [0,t0]with a compact orientable normalized three-manifold as initial data, where each δ-cutoﬀ at a time t∈[0,t0]has δ=δ(t)≤min{/tildewideδ(t),/tildewider(2t)}. Letr0,t0satisfyθ−1h≤r0≤¯r√t0andt0≥T, whereh is the maximal cutoﬀ radius for surgeries in [t0 2,t0] (if there is no surgery in the time interval [t0 2,t0], we takeh= 0),and assume that the solution on the ball Bt0(x0,r0) satisﬁes Rm(x,t0)≥ −r−2 0,onBt0(x0,r0), and Volt0(Bt0(x0,r0))≥wr3 0. Then the solution is well deﬁned and satisﬁes R(x,t)<Kr−2 0 in the whole parabolic neighborhood P/parenleftig x0,t0,r0 4,−τr2 0/parenrightig =/braceleftig (x,t)|x∈Bt/parenleftig x0,r0 4/parenrightig ,t∈[t0−τr2 0,t0]/bracerightig . Proof. We are given that Rm(x,t0)≥ −r−2 0forx∈Bt0(x0,r0), and Volt0(Bt0(x0,r0))≥wr3 0. The same argument in the derivation of (7.2.7) and (7.2.8) (by using the Alexandrov space theory) implies that there exists a ball Bt0(x′,r′)⊂Bt0(x0,r0) with (7.5.17) Vol t0(Bt0(x′,r′))≥1 2α3(r′)3 and with (7.5.18) r′≥c(w)r0 for some small positive constant c(w) depending only on w, whereα3is the volume of the unit ball in R3. As in (7.1.2), we can rewrite the pinching assumption (7.3.3 ) as Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R, where y=f(x) =x(logx−3),fore2≤x<+∞, 464 H.-D. CAO AND X.-P. ZHU is increasing and convex with range −e2≤y <+∞, and its inverse function is also increasing and satisﬁes lim y→+∞f−1(y)/y= 0. Note thatt0r−2 0≥¯r−2by the hypotheses. We may require T(w)≥8c(w)−1. Then by applying Theorem 7.5.1 (iii) with A= 8c(w)−1and combining with the pinching assumption, we can reduce the proof of the theorem to the spec ial casew=1 2α3. In the following we simply assume w=1 2α3. Let us ﬁrst consider the case r0</tildewider(t0). We claim that R(x,t0)≤C2 0r−2 0on Bt0(x0,r0 3), for some suﬃciently large positive constant C0depending only on ε. If not, then there is a canonical neighborhood around ( x,t0). Note that the type (c) canon- ical neighborhood has already been ruled out by our design of cutoﬀ surgeries. Thus (x,t0) belongs to an ε-neck or an ε-cap. This tells us that there is a nearby point y, withR(y,t0)≥C−1 2R(x,t0)>C−1 2C2 0r−2 0anddt0(y,x)≤C1R(x,t0)−1 2≤C1C−1 0r0, which is the center of the ε-neckBt0(y,ε−1R(y,t0)−1 2). (HereC1,C2are the pos- itive constants in the deﬁnition of canonical neighborhood assumption). Clearly, when we choose C0to be much larger than C1,C2andε−1, the whole ε-neck Bt0(y,ε−1R(y,t0)−1 2) is contained in Bt0(x0,r0 2) and we have (7.5.19)Volt0(Bt0(y,ε−1R(y,t0)−1 2)) (ε−1R(y,t0)−1 2)3≤8πε2. Without loss of generality, we may assume ε>0 is very small. Since we have assumed thatRm≥ −r−2 0onBt0(x0,r0) and Vol t0(Bt0(x0,r0))≥1 2α3r3 0, we then get a contradiction by applying the standard Bishop-Gromov volu me comparison. Thus we have the desired curvature estimate R(x,t0)≤C2 0r−2 0onBt0(x0,r0 3). Furthermore, by using the gradient estimates in the deﬁniti on of canonical neigh- borhood assumption, we can take K= 10C2 0,τ=1 100η−1C−2 0andθ=1 5C−1 0in this case. And since r0≥θ−1h, we haveR <10C2 0r−2 0≤1 2h−2and the surgeries do not interfere in P(x0,t0,r0 4,−τr2 0). We now consider the remaining case /tildewider(t0)≤r0≤¯r√t0. Let us redeﬁne τ= min/braceleftbigg¯τ0 2,1 100η−1C−2 0/bracerightbigg , K= max/braceleftbigg 2/parenleftbigg ¯C+2¯B ¯τ0/parenrightbigg ,25C2 0/bracerightbigg , and θ=1 2K−1 2 where ¯τ0=τ0(w),¯B=B(w) and ¯C=C(w) are the positive constants in Theorem 6.3.3(ii) with w=1 2α3, andC0is the positive constant chosen above. We will show there is a suﬃciently small ¯ r>0 such that the conclusion of the theorem for w=1 2α3 holds for the chosen τ,Kandθ. Argue by contradiction. Suppose not, then there exist a sequ ence of ¯rα→0, and a sequence of solutions gα ijwith points ( xα 0,tα 0) and radii rα 0such that the as- sumptions of the theorem do hold with /tildewider(tα 0)≤rα 0≤¯rα/radicalbigtα 0whereas the conclusion THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 465 does not. Similarly as in the proof of Theorem 7.2.2, we claim that we may as- sume that for all suﬃciently large α, any other point ( xα,tα) and radius rα>0 with that property has either tα> tα 0ortα=tα 0withrα≥rα 0; moreover tα tends to + ∞asα→+∞. Indeed, for ﬁxed αand the solution gα ij, lettα minbe the inﬁmum of all possible times tαwith some point xαand some radius rαhav- ing that property. Since each such tαsatisﬁes ¯rα√ tα≥rα≥/tildewider(tα), it follows that whenαis large,tα minmust be positive and very large. Clearly for each ﬁxed suﬃ- ciently large α, by passing to a limit, there exist some point xα minand some radius rα min(≥/tildewider(tα min)>0) so that all assumptions of the theorem still hold for ( xα min,tα min) andrα min, whereas the conclusion of the theorem does not hold with R≥K(rα min)−2 somewhere in P(xα min,tα min,1 4rα min,−τ(rα min)2) for all suﬃciently large α. Here we used the fact that if R < K (rα min)−2onP(xα min,tα min,1 4rα min,−τ(rα min)2), then there is noδ-cutoﬀ surgery there; otherwise there must be a point there w ith scalar cur- vature at least1 2δ−2(tα min 2)(/tildewider(tα min 2))−2≥1 2(/tildewider(tα min))−2(/tildewider(tα min 2))−2≫K(rα min)−2since ¯rα/radicalbigtα min≥rα min≥/tildewider(tα min) and ¯rα→0, which is a contradiction. After choosing the ﬁrst time tα min, by passing to a limit again, we can then choose rα minto be the smallest radius for all possible ( xα min,tα min)’s andrα min’s with that prop- erty. Thus we have veriﬁed the claim. For simplicity, we will drop the index αin the following arguments. By the assumption and the standard volume comparison, we have Volt0/parenleftbigg Bt0/parenleftbigg x0,1 2r0/parenrightbigg/parenrightbigg ≥ξ0r3 0 for some universal positive ξ0. As in deriving (7.2.7) and (7.2.8), we can ﬁnd a ball Bt0(x′ 0,r′ 0)⊂Bt0(x0,r0 2) with Volt0(Bt0(x′ 0,r′ 0))≥1 2α3(r′ 0)3and1 2r0≥r′ 0≥ξ′ 0r0 for some universal positive constant ξ′ 0. Then by what we had proved in the previous case and by the choice of the ﬁrst time t0and the smallest radius r0, we know that the solution is deﬁned in P(x′ 0,t0,r′ 0 4,−τ(r′ 0)2) with the curvature bound R<K (r′ 0)−2≤K(ξ′ 0)−2r−2 0. Since ¯r√t0≥r0≥/tildewider(t0) and ¯r→0 asα→ ∞, we see that t0→+∞andt0r−2 0→+∞ asα→+∞. DeﬁneT(w) = 8c(w)−1+¯ξ, for some suitable large universal positive constant ¯ξ. Then for αsuﬃciently large, we can apply Theorem 7.5.1(iii) and the pinching assumption to conclude that (7.5.20) R≤K′r−2 0,onP/parenleftig x0,t0,4r0,−τ 2(ξ′ 0)2r2 0/parenrightig , for some positive constant K′depending only on Kandξ′ 0. Furthermore, by combining with the pinching assumption, we deduce that when αsuﬃciently large, Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R (7.5.21) ≥ −r−2 0 466 H.-D. CAO AND X.-P. ZHU onP(x0,t0,r0,−τ 2(ξ′ 0)2r2 0). So by applying Theorem 6.3.3(ii) with w=1 2α3, we have that whenαsuﬃciently large, (7.5.22) Vol t(Bt(x0,r0))≥ξ1r3 0, for allt∈[t0−τ 2(ξ′ 0)2r2 0,t0], and (7.5.23) R≤/parenleftbigg ¯C+2¯B ¯τ0/parenrightbigg r−2 0≤1 2Kr−2 0 onP(x0,t0,r0 4,−τ 2(ξ′ 0)2r2 0), whereξ1is some universal positive constant. Next we want to extend the estimate (7.5.23) backwards in tim e. Denote by t1=t0−τ 2(ξ′ 0)2r2 0. The estimate (7.5.22) gives Volt1(Bt1(x0,r0))≥ξ1r3 0. By the same argument in the derivation of (7.2.7) and (7.2.8) again, we can ﬁnd a ballBt1(x1,r1)⊂Bt1(x0,r0) with Volt1(Bt1(x1,r1))≥1 2α3r3 1 and with r1≥ξ′ 1r0 for some universal positive constant ξ′ 1. Then by what we had proved in the previous case and by the lower bound (7.5.21) at t1and the choice of the ﬁrst time t0, we know that the solution is deﬁned on P(x1,t1,r1 4,−τr2 1) with the curvature bound R < Kr−2 1. By applying Theorem 7.5.1(iii) and the pinching assumptio n again we get that for αsuﬃciently large, (7.5.20)′R≤K′′r−2 0 onP(x0,t1,4r0,−τ 2(ξ′ 1)2r2 0), for some positive constant K′′depending only on Kand ξ′ 1. Moreover, by combining with the pinching assumption, we ha ve Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R (7.5.21)′ ≥ −r−2 0 onP(x0,t1,4r0,−τ 2(ξ′ 1)2r2 0), forαsuﬃciently large. So by applying Theorem 6.3.3 (ii) withw=1 2α3again, we have that for αsuﬃciently large, (7.5.22)′Volt(Bt(x0,r0))≥ξ1r3 0, for allt∈[t0−τ 2(ξ′ 0)2r2 0−τ 2(ξ′ 1)2r2 0,t0], and (7.5.23)′R≤/parenleftbigg ¯C+2¯B ¯τ0/parenrightbigg r−2 0≤1 2Kr−2 0 onP(x0,t0,r0 4,−τ 2(ξ′ 0)2r2 0−τ 2(ξ′ 1)2r2 0). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 467 Note that the constants ξ0,ξ′ 0,ξ1andξ′ 1are universal, independent of the time t1and the choice of the ball Bt1(x1,r1). Then we can repeat the above procedure as many times as we like, until we reach the time t0−τr2 0. Hence we obtain the estimate (7.5.23)′′R≤(¯C+2¯B ¯τ0)r−2 0≤1 2Kr−2 0 onP(x0,t0,r0 4,−τr2 0), for suﬃciently large α. This contradicts the choice of the point (x0,t0) and the radius r0which make R≥Kr−2 0somewhere in P(x0,t0,r0 4,−τr2 0). Therefore we have completed the proof of the theorem. Consequently, we have the following result which is analog o f Corollary 7.2.4. Corollary 7.5.3. For anyε >0andw >0, there exist ¯r= ¯r(w,ε)>0, θ=θ(w,ε)>0andT=T(w)with the following property. Suppose we have a solution, constructed by Theorem 7.4.3with the positive functions /tildewideδ(t)and/tildewider(t), to the Ricci ﬂow with surgery with a compact orientable normali zed three-manifold as initial data, where each δ-cutoﬀ at a time thasδ=δ(t)≤min{/tildewideδ(t),/tildewider(2t)}. If Bt0(x0,r0)is a geodesic ball at time t0, withθ−1h≤r0≤¯r√t0andt0≥T, whereh is the maximal cutoﬀ radii for surgeries in [t0 2,t0](if there is no surgery in the time interval [t0 2,t0], we takeh= 0), and satisﬁes min{Rm(x,t0)|x∈Bt0(x0,r0)}=−r−2 0, then Volt0(Bt0(x0,r0))<wr3 0. Proof. We argue by contradiction. Let θ=θ(w,ε) andT= 2T(w), whereθ(w,ε) andT(w) are the positive constant in Theorem 7.5.2. Suppose for any ¯r>0 there is a solution and a geodesic ball Bt0(x0,r0) satisfying the assumptions of the corollary withθ−1h≤r0≤¯r√t0andt0≥T, and with min{Rm(x,t0)|x∈Bt0(x0,r0)}=−r−2 0, but Volt0(Bt0(x0,r0))≥wr3 0. Without loss of generality, we may assume that ¯ ris less than the corresponding constant in Theorem 7.5.2. We can then apply Theorem 7.5.2 to get R(x,t)≤Kr−2 0 whenevert∈[t0−τr2 0,t0] anddt(x,x0)≤r0 4, whereτandKare the positive constants in Theorem 7.5.2. Note that t0r−2 0≥¯r−2→+∞as ¯r→0. By combining with the pinching assumption we have Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R ≥ −1 2r−2 0 468 H.-D. CAO AND X.-P. ZHU in the region P(x0,t0,r0 4,−τr2 0) ={(x,t)|x∈Bt(x0,r0 4),t∈[t0−τr2 0,t0]}, provided ¯r>0 is suﬃciently small. Thus we get the estimate |Rm| ≤K′r−2 0 inP(x0,t0,r0 4,−τr2 0), whereK′is a positive constant depending only on wandε. We can now apply Theorem 7.5.1 (iii) to conclude that R(x,t)≤/tildewideKr−2 0 whenevert∈[t0−τ 2r2 0,t0] anddt(x,x0)≤r0, where/tildewideKis a positive constant depending only onwandε. By using the pinching assumption again we further have Rm(x,t)≥ −1 2r−2 0 in the region P(x0,t0,r0,−τ 2r2 0) ={(x,t)|x∈Bt(x0,r0),t∈[t0−τ 2r2 0,t0]}, as long as ¯ris suﬃciently small. In particular, this would imply min{Rm(x,t0)|x∈Bt0(x0,r0)}>−r−2 0, which is a contradiction. Remark 7.5.4. In section 7.3 of [104], Perelman claimed a stronger stateme nt than the above Corollary 7.5.3 that allows r0<θ−1hin the assumptions. Neverthe- less, the above weaker statement is suﬃcient to deduce the ge ometrization result. 7.6. Long Time Behavior. In Section 5.3, we obtained the long time behavior for smooth (compact) solutions to the three-dimensional Ri cci ﬂow with bounded normalized curvature. The purpose of this section is to adap t the arguments there to solutions of the Ricci ﬂow with surgery and to drop the bounde d normalized curvature assumption. Recall from Corollary 7.4.4 that we have completely underst ood the topological structure of a compact, orientable three-manifold with non negative scalar curvature. From now on we assume that our initial manifold does not admit any metric with nonnegative scalar curvature, and that once we get a compact component with nonnegative scalar curvature, it is immediately removed . Furthermore, if a solution to the Ricci ﬂow with surgery becomes extinct in a ﬁn ite time, we have also obtained the topological structure of the initial manifold . So in the following we only consider those solutions to the Ricci ﬂow with surgery which exist for all times t≥0. Letgij(t), 0≤t <+∞, be a solution to the Ricci ﬂow with δ-cutoﬀ surgeries, constructed by Theorem 7.4.3 with normalized initial data. Let 0<t1<t2<···< tk<···be the surgery times, where each δ-cutoﬀ at a time tkhasδ=δ(tk)≤ min{/tildewideδ(tk),/tildewider(2tk)}. On each time interval ( tk−1,tk) (denote by t0= 0), the scalar curvature satisﬁes the evolution equation (7.6.1)∂ ∂tR= ∆R+ 2|◦ Ric|2+2 3R2 where◦ Ric is the trace-free part of Ric .ThenRmin(t), the minimum of the scalar curvature at the time t, satisﬁes d dtRmin(t)≥2 3R2 min(t) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 469 fort∈(tk−1,tk), for eachk= 1,2,.... Since our surgery procedure had removed all components with nonnegative scalar curvature, the minimum Rmin(t) is negative for allt∈[0,+∞). Also recall that the cutoﬀ surgeries were performed only o nδ-necks. Thus the surgeries do not occur at the parts where Rmin(t) are achieved. So the diﬀerential inequality d dtRmin(t)≥2 3R2 min(t) holds for all t≥0, and then by normalization, Rmin(0)≥ −1, we have (7.6.2) Rmin(t)≥ −3 2·1 t+3 2,for allt≥0. Meanwhile, on each time interval ( tk−1,tk), the volume satisﬁes the evolution equation d dtV=−/integraldisplay RdV and then by (7.6.2), d dtV≤3 2·1 (t+3 2)V. Since the cutoﬀ surgeries do not increase volume, we thus hav e (7.6.3)d dtlog/parenleftigg V(t)/parenleftbigg t+3 2/parenrightbigg−3 2/parenrightigg ≤0 for allt≥0. Equivalently, the function V(t)(t+3 2)−3 2is nonincreasing on [0 ,+∞). We can now use the monotonicity of the function V(t)(t+3 2)−3 2to extract the information of the solution at large times. On each time inte rval (tk−1,tk), we have d dtlog/parenleftigg V(t)/parenleftbigg t+3 2/parenrightbigg−3 2/parenrightigg =−/parenleftigg Rmin(t) +3 2/parenleftbig t+3 2/parenrightbig/parenrightigg +1 V/integraldisplay M(Rmin(t)−R)dV. Then by noting that the cutoﬀ surgeries do not increase volum e, we get V(t) (t+3 2)3 2≤V(0) (3 2)3 2exp/braceleftigg −/integraldisplayt 0/parenleftbigg Rmin(t) +3 2(t+3 2)/parenrightbigg dt (7.6.4) −/integraldisplayt 01 V/integraldisplay M(R−Rmin(t))dVdt/bracerightigg for allt>0. Now by this inequality and the equation (7.6.1), we obtain the following consequence. Lemma 7.6.1. Letgij(t)be a solution to the Ricci ﬂow with surgery, constructed by Theorem 7.4.3with normalized initial data. If for a ﬁxed 0< r < 1and a se- quence of times tα→ ∞, the rescalings of the solution on the parabolic neighborho ods 470 H.-D. CAO AND X.-P. ZHU P(xα,tα,r√ tα,−r2tα) ={(x,t)|x∈Bt(xα,r√ tα),t∈[tα−r2tα,tα]}, with factor (tα)−1and shifting the times tαto1, converge in the C∞topology to some smooth lim- iting solution, deﬁned in an abstract parabolic neighborho odP(¯x,1,r,−r2), then this limiting solution has constant sectional curvature −1/4tat any time t∈[1−r2,1]. In the previous section we obtained several curvature estim ates for the solutions to the Ricci ﬂow with surgery. Now we combine the curvature es timates with the above lemma to derive the following asymptotic result for th e curvature. Lemma 7.6.2 ( Perelman [104] ).For anyε >0, letgij(t),0≤t <+∞, be a solution to the Ricci ﬂow with surgery, constructed by Theor em7.4.3with normalized initial data. (i) Givenw >0,r >0,ξ >0, one can ﬁnd T=T(w,r,ξ,ε )<+∞such that if the geodesic ball Bt0(x0,r√t0)at some time t0≥Thas volume at least wr3t3 2 0and the sectional curvature at least −r−2t−1 0, then the curvature at x0 at timet=t0satisﬁes (7.6.5) |2tRij+gij|<ξ. (ii) Given in addition 1≤A<∞and allowing Tto depend on A, we can ensure (7.6.5)for all points in Bt0(x0,Ar√t0). (iii) The same is true for all points in the forward parabolic neighborhood P(x0,t0,Ar√t0,Ar2t0)/defines{(x,t)|x∈Bt(x0,Ar√t0),t∈[t0,t0+Ar2t0]}. Proof. (i) By the assumptions and the standard volume comparison, w e have Volt0(Bt0(x0,ρ))≥cwρ3 for all 0< ρ≤r√t0, wherecis a universal positive constant. Let ¯ r= ¯r(cw,ε) be the positive constant in Theorem 7.5.2 and set r0= min {r,¯r}. OnBt0(x0,r0√t0)(⊂ Bt0(x0,r√t0)), we have Rm≥ −(r0√t0)−2(7.6.6) and Vol t0(Bt0(x0,r0√t0))≥cw(r0√t0)3. Obviously, there holds θ−1h≤r0√t0≤¯r√t0whent0is large enough, where θ= θ(cw,ε) is the positive constant in Theorem 7.5.2 and his the maximal cutoﬀ radius for surgeries in [t0 2,t0] (if there is no surgery in the time interval [t0 2,t0], we take h= 0). Then it follows from Theorem 7.5.2 that the solution is d eﬁned and satisﬁes R<K (r0√t0)−2 on whole parabolic neighborhood P(x0,t0,r0√t0 4,−τ(r0√t0)2). Hereτ=τ(cw,ε) and K=K(cw,ε) are the positive constants in Theorem 7.5.2. By combining w ith the pinching assumption we have Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R ≥ −const.K(r0√t0)−2 in the region P(x0,t0,r0√t0 4,−τ(r0√t0)2). Thus we get the estimate (7.6.7) |Rm| ≤K′(r0√t0)−2 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 471 onP(x0,t0,r0√t0 4,−τ(r0√t0)2), for some positive constant K′=K′(w,ε) depending only onwandε. The curvature estimate (7.6.7) and the volume estimate (7.6 .6) ensure that as t0→+∞we can take smooth (subsequent) limits for the rescalings of the solution with factor ( t0)−1on parabolic neighborhoods P(x0,t0,r0√t0 4,−τ(r0√t0)2). Then by applying Lemma 7.6.1, we can ﬁnd T=T(w,r,ξ,ε )<+∞such that when t0≥T, there holds (7.6.8) |2tRij+gij|<ξ, onP(x0,t0,r0√t0 4,−τ(r0√t0)2), in particular, |2tRij+gij|(x0,t0)<ξ. This proves the assertion (i). (ii) In view of the above argument, to get the estimate (7.6.5 ) for all points inBt0(x0,Ar√t0), the key point is to get a upper bound for the scalar curvatur e on the parabolic neighborhood P(x0,t0,Ar√t0,−τ(r0√t0)2). After having the estimates (7.6.6) and (7.6.7), one would like to use Theorem 7.5.1(iii ) to obtain the desired scalar curvature estimate. Unfortunately it does not work since ou rr0may be much larger than the constant ¯ r(A,ε) there when Ais very large. In the following we will use Theorem 7.5.1(ii) to overcome the diﬃculty. Given 1 ≤A<+∞, based on (7.6.6) and (7.6.7), we can use Theorem 7.5.1(ii) t o ﬁnd a positive constant K1=K1(w,r,A,ε ) such that each point in Bt0(x0,2Ar√t0) with its scalar curvature at least K1(r√t0)−2has a canonical neighborhood. We claim that there exists T=T(w,r,A,ε )<+∞so that when t0≥T, we have (7.6.9) R<K 1(r√t0)−2,onBt0(x0,2Ar√t0). Argue by contradiction. Suppose not; then there exist a sequ ence of times tα 0→ +∞and sequences of points xα 0,xαwithxα∈Btα 0(xα 0,2Ar/radicalbigtα 0) andR(xα,tα 0) = K1(r/radicalbigtα 0)−2. Since there exist canonical neighborhoods ( ε-necks orε-caps) around the points ( xα,tα 0), there exist positive constants c1,C2depending only on εsuch that Voltα 0(Btα 0(xα,K−1 2 1(r/radicalbig tα 0)))≥c1(K−1 2 1(r/radicalbig tα 0))3 and C−1 2K1(r/radicalbig tα 0)−2≤R(x,tα 0)≤C2K1(r/radicalbig tα 0)−2, onBtα 0(xα,K−1 2 1(r/radicalbigtα 0)), for allα. By combining with the pinching assumption we have Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R ≥ −const.C2K1(r/radicalbig tα 0)−2, onBtα 0(xα,K−1 2 1(r/radicalbigtα 0)), for allα. It then follows from the assertion (i) we just proved that lim α→+∞|2tRij+gij|(xα,tα 0) = 0. 472 H.-D. CAO AND X.-P. ZHU In particular, we have tα 0R(xα,tα 0)<−1 forαsuﬃciently large. This contradicts our assumption that R(xα,tα 0) = K1(r/radicalbigtα 0)−2. So we have proved assertion (7.6.9). Now by combining (7.6.9) with the pinching assumption as bef ore, we have (7.6.10) Rm≥ −K2(r√t0)−2 onBt0(x0,2Ar√t0), whereK2=K2(w,r,A,ε ) is some positive constant depending only onw,r,Aandε. Thus by (7.6.9) and (7.6.10) we have (7.6.11) |Rm| ≤K′ 1(r√t0)−2,onBt0(x0,2Ar√t0), for some positive constant K′ 1=K′ 1(w,r,A,ε ) depending only on w,r,Aandε. This gives us the curvature estimate on Bt0(x0,2Ar√t0) for allt0≥T(w,r,A,ε ). From the arguments in proving the above assertion (i), we hav e the estimates (7.6.6) and (7.6.7) and the solution is well-deﬁned on the wh ole parabolic neighbor- hoodP(x0,t0,r0√t0 4,−τ(r0√t0)2) for allt0≥T(w,r,A,ε ). Clearly we may assume that (K′ 1)−1 2r <min{r0 4,√τr0}. Thus by combining with the curvature estimate (7.6.11), we can apply Theorem 7.5.1(i) to get the following volume control (7.6.12) Vol t0(Bt0(x,(K′ 1)−1 2r√t0))≥κ((K′ 1)−1 2r√t0)3 for anyx∈Bt0(x0,Ar√t0), whereκ=κ(w,r,A,ε ) is some positive constant depend- ing only on w,r,Aandε. So by using the assertion (i), we see that for t0≥Twith T=T(w,r,ξ,A,ε ) large enough, the curvature estimate (7.6.5) holds for all points in Bt0(x0,Ar√t0). (iii) We next want to extend the curvature estimate (7.6.5) t o all points in the forward parabolic neighborhood P(x0,t0,Ar√t0,Ar2t0). Consider the time interval [t0,t0+Ar2t0] in the parabolic neighborhood. In assertion (ii), we have o btained the desired estimate (7.6.5) at t=t0. Suppose estimate (7.6.5) holds on a maximal time interval [t0,t′) witht′≤t0+Ar2t0. This says that we have (7.6.13) |2tRij+gij|<ξ onP(x0,t0,Ar√t0,t′−t0)/defines{(x,t)|x∈Bt(x0,Ar√t0),t∈[t0,t′)}so that either there exists a surgery in the ball Bt′(x0,Ar√t0) att=t′, or there holds |2tRij+gij|= ξsomewhere in Bt′(x0,Ar√t0) att=t′. Since the Ricci curvature is near −1 2t′in the geodesic ball, the surgeries cannot occur there. Thus we onl y need to consider the latter possibility. Recall that the evolution of the length of a curve γand the volume of a domain Ω are given by d dtLt(γ) =−/integraldisplay γRic (˙γ,˙γ)dst andd dtVolt(Ω) = −/integraldisplay ΩRdV t. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 473 By substituting the curvature estimate (7.6.13) into the ab ove two evolution equations and using the volume lower bound (7.6.6), it is not hard to see (7.6.14) Vol t′(Bt′(x0,√ t′))≥κ′(t′)3 2 for some positive constant κ′=κ′(w,r,ξ,A,ε ) depending only on w,r,ξ,Aandε. Then by the above assertion (ii), the combination of the curv ature estimate (7.6.13) and the volume lower bound (7.6.14) implies that the curvatu re estimate (7.6.5) still holds for all points in Bt′(x0,Ar√t0) provided T=T(w,r,ξ,A,ε ) is chosen large enough. This is a contradiction. Therefore we have proved as sertion (iii). We now state and prove the important Thick-thin decomposition theorem . A more general version (without the restriction on ε) was implicitly claimed by Perelman in [103] and [104]. Theorem 7.6.3 ( The Thick-thin decomposition theorem ).For anyw >0and 0< ε≤1 2w, there exists a positive constant ρ=ρ(w,ε)≤1with the following property. Suppose gij(t) (t∈[0,+∞))is a solution, constructed by Theorem 7.4.3with the nonincreasing (continuous )positive functions /tildewideδ(t)and/tildewider(t), to the Ricci ﬂow with surgery and with a compact orientable normalized three-man ifold as initial data, where eachδ-cutoﬀ at a time thasδ=δ(t)≤min{/tildewideδ(t),/tildewider(2t)}. Then for any arbitrarily ﬁxedξ >0, fortlarge enough, the manifold Mtat timetadmits a decomposition Mt=Mthin(w,t)∪Mthick(w,t)with the following properties: (a) For every x∈Mthin(w,t), there exists some r=r(x,t)>0, with 0<r√ t< ρ√ t, such that Rm≥ −(r√ t)−2on B t(x,r√ t),and Volt(Bt(x,r√ t))<w(r√ t)3. (b) For every x∈Mthick(w,t), we have |2tRij+gij|<ξ on B t(x,ρ√ t),and Volt(Bt(x,ρ√ t))≥1 10w(ρ√ t)3. Moreover, if we take any sequence of points xα∈Mthick(w,tα),tα→+∞, then the scalings of gij(tα)aroundxαwith factor (tα)−1converge smoothly, along a subse- quence ofα→+∞, to a complete hyperbolic manifold of ﬁnite volume with cons tant sectional curvature −1 4. Proof. Let ¯r= ¯r(w,ε),θ=θ(w,ε) andhbe the positive constants obtained in Corollary 7.5.3. We may assume ρ≤¯r≤e−3. For any point x∈Mt, there are two cases: either (i) min {Rm|Bt(x,ρ√ t)} ≥ − (ρ√ t)−2, or (ii) min {Rm|Bt(x,ρ√ t)}<−(ρ√ t)−2. 474 H.-D. CAO AND X.-P. ZHU Let us ﬁrst consider Case (i). If Vol t(Bt(x,ρ√ t))<1 10w(ρ√ t)3, then we can chooserslightly less than ρso that Rm≥ −(ρ√ t)−2≥ −(r√ t)−2 onBt(x,r√ t)(⊂Bt(x,ρ√ t)), and Volt(Bt(x,r√ t))<1 10w(ρ√ t)3<w(r√ t)3; thusx∈Mthin(w,t). If Vol t(Bt(x,ρ√ t))≥1 10w(ρ√ t)3, we can apply Lemma 7.6.2(ii) to conclude that for tlarge enough, |2tRij+gij|<ξ onBt(x,ρ√ t); thusx∈Mthick(w,t). Next we consider Case (ii). By continuity, there exists 0 < r=r(x,t)< ρsuch that (7.6.15) min {Rm|Bt(x,r√ t)}=−(r√ t)−2. Ifθ−1h≤r√ t(≤¯r√ t), we can apply Corollary 7.5.3 to conclude Volt(Bt(x,r√ t))<w(r√ t)3; thusx∈Mthin(w,t). We now consider the diﬃcult subcase r√ t<θ−1h. By the pinching assumption, we have R≥(r√ t)−2(log[(r√ t)−2(1 +t)]−3) ≥(logr−2−3)(r√ t)−2 ≥2(r√ t)−2 ≥2θ2h−2 somewhere in Bt(x,r√ t). Sincehis the maximal cutoﬀ radius for surgeries in [t 2,t], by the design of the δ-cutoﬀ surgery, we have h≤sup/braceleftbigg δ2(s)/tildewider(s)|s∈/bracketleftbiggt 2,t/bracketrightbigg/bracerightbigg ≤/tildewideδ/parenleftbiggt 2/parenrightbigg /tildewider(t)/tildewider/parenleftbiggt 2/parenrightbigg . Note also/tildewideδ(t 2)→0 ast→+∞. Thus from the canonical neighborhood assumption, we see that for tlarge enough, there exists a point in the ball Bt(x,r√ t) which has a canonical neighborhood. We claim that for tsuﬃciently large, the point xsatisﬁes R(x,t)≥1 2(r√ t)−2, and then the above argument shows that the point xalso has a canonical ε-neck or ε-cap neighborhood. Otherwise, by continuity, we can choose a pointx∗∈Bt(x,r√ t) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 475 withR(x∗,t) =1 2(r√ t)−2. Clearly the new point x∗has a canonical neighborhood B∗by the above argument. In particular, there holds C−1 2(ε)R≤1 2(r√ t)−2≤C2(ε)R on the canonical neighborhood B∗. By the deﬁnition of canonical neighborhood as- sumption, we have Bt(x∗,σ∗)⊂B∗⊂Bt(x∗,2σ∗) for someσ∗∈(0,C1(ε)R−1 2(x∗,t)). Clearly, without loss of generality, we may assume (in the deﬁnition of canonical neighborhood assumption) th atσ∗>2R−1 2(x∗,t).Then R(1 +t)≥1 2C−1 2(ε)r−2 ≥1 2C−1 2(ε)ρ−2 onBt(x∗,2r√ t).Thus when we choose ρ=ρ(w,ε) small enough, it follows from the pinching assumption that Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R ≥ −1 2(r√ t)−2, onBt(x∗,2r√ t).This is a contradiction with (7.6.15). We have seen that tR(x,t)≥1 2r−2(≥1 2ρ−2). Sincer−2>θ2h−2tin this subcase, we conclude that for arbitrarily given A<+∞, (7.6.16) tR(x,t)>A2ρ−2 as long astis large enough. LetB, withBt(x,σ)⊂B⊂Bt(x,2σ), be the canonical ε-neck orε-cap neigh- borhood of ( x,t). By the deﬁnition of the canonical neighborhood assumptio n, we have 0<σ<C 1(ε)R−1 2(x,t), C−1 2(ε)R≤R(x,t)≤C2(ε)R,onB, and (7.6.17) Vol t(B)≤εσ3≤1 2wσ3. Choose 0<A<C 1(ε) so thatσ=AR−1 2(x,t). For suﬃciently large t, since R(1 +t)≥C−1 2(ε)(tR(x,t)) ≥1 2C−1 2(ε)ρ−2, 476 H.-D. CAO AND X.-P. ZHU onB, we can require ρ=ρ(w,ε) to be smaller still, and use the pinching assumption to conclude Rm≥ −[f−1(R(1 +t))/(R(1 +t))]R (7.6.18) ≥ −(AR−1 2(x,t))−2 =−σ−2, onB. For suﬃciently large t, we adjust r=σ(√ t)−1(7.6.19) = (AR−1 2(x,t))(√ t)−1 <ρ, by (7.6.16). Then the combination of (7.6.17), (7.6.18) and (7.6.19) implies that x∈Mthin(w,t). The last statement in (b) follows directly from Lemma 7.6.2. (Here we also used Bishop-Gromov volume comparison, Theorem 7.5.2 and Ha milton’s compactness theorem to take a subsequent limit.) Therefore we have completed the proof of the theorem. To state the long-time behavior of a solution to the Ricci ﬂow with surgery, we ﬁrst recall some basic terminology in three-dimensional to pology. A three-manifold Mis called irreducible if every smooth two-sphere embedded in Mbounds a three- ball inM. If we have a solution ( Mt,gij(t)) obtained by Theorem 7.4.3 with a compact, orientable and irreducible three-manifold ( M,g ij) as initial data, then at each timet >0, by the cutoﬀ surgery procedure, the solution manifold Mtconsists of a ﬁnite number of components where one of the components, c alled the essential component and denoted by M(1) t, is diﬀeomorphic to the initial manifold Mwhile the rest are diﬀeomorphic to the three-sphere S3. The main result of this section is the following generalizat ion of Theorem 5.3.4. A more general version of the result (without the restriction onε) was implicitly claimed by Perelman in [104]. Theorem 7.6.4 ( Long-time behavior of the Ricci ﬂow with surgery ).Letw> 0and0< ε≤1 2wbe any small positive constants and let (Mt,gij(t)),0< t < +∞,be a solution to the Ricci ﬂow with surgery, constructed by Th eorem 7.4.3with the nonincreasing (continuous )positive functions /tildewideδ(t)and/tildewider(t)and with a compact, orientable, irreducible and normalized three-manifold Mas initial data, where each δ-cutoﬀ at a time thasδ=δ(t)≤min{/tildewideδ(t),/tildewider(2t)}. Then one of the following holds: either (i) for all suﬃciently large t, we haveMt=Mthin(w,t); or (ii) there exists a sequence of times tα→+∞such that the scalings of gij(tα)on the essential component M(1) tα, with factor (tα)−1, converge in the C∞topol- ogy to a hyperbolic metric on the initial compact manifold Mwith constant sectional curvature −1 4; or (iii) we can ﬁnd a ﬁnite collection of complete noncompact hy perbolic three- manifolds H1,...,Hm, with ﬁnite volume, and compact subsets K1,...,K m ofH1,...,Hmrespectively obtained by truncating each cusp of the hyperb olic manifolds along constant mean curvature torus of small area , and for all t beyond some time T <+∞we can ﬁnd diﬀeomorphisms ϕl,1≤l≤m, ofKl THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 477 intoMtso that as long as tis suﬃciently large, the metric t−1ϕ∗ l(t)gij(t)is as close to the hyperbolic metric as we like on the compact set sK1,...,K m; moreover, the complement Mt\(ϕ1(K1)∪ ··· ∪ϕm(Km))is contained in the thin partMthin(w,t), and the boundary tori of each Klare incompressible in the sense that each ϕlinjectsπ1(∂Kl)intoπ1(Mt). Proof. The proof of the theorem follows, with some modiﬁcations, es sentially from the same argument of Hamilton as in the proof of Theorem 5 .3.4. Clearly we may assume that the thick part Mthick(w,t) is not empty for a sequence tα→+∞, since otherwise we have case (i). If we take a sequence of poi ntsxα∈ Mthick(w,tα), then by Theorem 7.6.3(b) the scalings of gij(tα) aroundxαwith factor (tα)−1converge smoothly, along a subsequence of α→+∞, to a complete hyperbolic manifold of ﬁnite volume with constant sectional curvature −1 4. The limits may be diﬀerent for diﬀerent choices of ( xα,tα). If a limit is compact, we have case (ii). Thus we assume that all limits are noncompact. Consider all the possible hyperbolic limits of the solution , and among them choose one such complete noncompact hyperbolic three-manifold Hwith the least possible number of cusps. Denote by hijthe hyperbolic metric of H. For all small a >0 we can truncate each cusp of Halong a constant mean curvature torus of area awhich is uniquely determined; we denote the remainder by Ha. Fixa >0 so small that Lemma 5.3.7 is applicable for the compact set K=Ha. Pick an integer l0suﬃciently large and an ǫ0suﬃciently small to guarantee from Lemma 5.3.8 that the iden tity mapIdis the only harmonic map FfromHato itself with taking ∂Hato itself, with the normal derivative of Fat the boundary of the domain normal to the boundary of the target, and with dCl0(Ha)(F,Id)< ǫ0. Then choose a positive integer q0and a small number δ0>0 from Lemma 5.3.7 such that if /tildewideFis a diﬀeomorphism of Ha into another complete noncompact hyperbolic three-manifo ld (/tildewideH,/tildewidehij) with no fewer cusps (than H), of ﬁnite volume and satisfying /ba∇dbl/tildewideF∗/tildewidehij−hij/ba∇dblCq0(Ha)≤δ0, then there exists an isometry IofHto/tildewideHsuch that dCl0(Ha)(/tildewideF,I)<ǫ0. By Lemma 5.3.8 we further require q0andδ0to guarantee the existence of a har- monic diﬀeomorphism from ( Ha,/tildewidegij) to ( Ha,hij) for any metric /tildewidegijonHawith /ba∇dbl/tildewidegij−hij/ba∇dblCq0(Ha)≤δ0. Letxα∈Mthick(w,tα),tα→+∞, be a sequence of points such that the scalings ofgij(tα) aroundxαwith factor ( tα)−1converge to hij. Then there exist a marked pointx∞∈ H aand a sequence of diﬀeomorphisms FαfromHaintoMtαsuch that Fα(x∞) =xαand /ba∇dbl(tα)−1F∗ αgij(tα)−hij/ba∇dblCm(Ha)→0 asα→ ∞ for all positive integers m. By applying Lemma 5.3.8 and the implicit function theorem, we can change Fαby an amount which goes to zero as α→ ∞ so as to make Fαa harmonic diﬀeomorphism taking ∂Hato a constant mean curvature hypersurface Fα(∂Ha) of (Mtα,(tα)−1gij(tα)) with the area aand satisfying the free boundary condition that the normal derivative of Fαat the boundary of the domain is normal to the boundary of the target; and by combining with Lemma 7.6.2 (iii), 478 H.-D. CAO AND X.-P. ZHU we can smoothly continue each harmonic diﬀeomorphism Fαforward in time a little to a family of harmonic diﬀeomorphisms Fα(t) from HaintoMtwith the metric t−1gij(t), withFα(tα) =Fαand with the time tslightly larger than tα, whereFα(t) takes∂Hainto a constant mean curvature hypersurface of ( Mt,t−1gij(t)) with the areaaand also satisﬁes the free boundary condition. Moreover, si nce the surgeries do not take place at the points where the scalar curvature is n egative, by the same argument as in Theorem 5.3.4 for an arbitrarily given positi ve integerq≥q0, positive numberδ < δ 0, and suﬃciently large α, we can ensure the extension Fα(t) satisﬁes /ba∇dblt−1F∗ α(t)gij(t)−hij/ba∇dblCq(Ha)≤δon a maximal time interval tα≤t≤ωα(ortα≤ t < ωαwhenωα= +∞), and with /ba∇dbl(ωα)−1F∗ α(ωα)gij(ωα)−hij/ba∇dblCq(Ha)=δ, when ωα<+∞. Here we have implicitly used the fact that Fα(ωα)(∂Ha) is still strictly concave to ensure the map Fα(ωα) is diﬀeomorphic. We further claim that there must be some αsuch thatωα= +∞; in other words, at least one hyperbolic piece persists. Indeed, suppose tha t for each large enough α we can only continue the family Fα(t) on a ﬁnite interval tα≤t≤ωα<+∞with /ba∇dbl(ωα)−1F∗ α(ωα)gij(ωα)−hij/ba∇dblCq(Ha)=δ. Consider the new sequence of manifolds ( Mωα,gij(ωα)). Clearly by Lemma 7.6.1, the scalings ofgij(ωα) around the new origins Fα(ωα)(x∞) with factor ( ωα)−1converge smoothly (by passing to a subsequence) to a complete noncomp act hyperbolic three- manifold/tildewideHwith the metric /tildewidehijand the origin /tildewidex∞and with ﬁnite volume. By the choice of the old limit H, the new limit /tildewideHhas at least as many cusps as H. By the deﬁnition of convergence, we can ﬁnd a sequence of compact su bsets/tildewideUαexhausting /tildewideHand containing /tildewidex∞, and a sequence of diﬀeomorphisms /tildewideFαof neighborhood of /tildewideUα intoMωαwith/tildewideFα(/tildewidex∞) =Fα(ωα)(x∞) such that for each compact subset /tildewideUof/tildewideHand each integer m, /ba∇dbl(ωα)−1/tildewideF∗ α(gij(ωα))−/tildewidehij/ba∇dblCm( /CTU)→0 asα→+∞. Thus for suﬃciently large α, we have the map Gα=/tildewideF−1 α◦Fα(ωα) :Ha→/tildewideH such that /ba∇dblG∗ α/tildewidehij−hij/ba∇dblCq(Ha)</tildewideδ for any ﬁxed /tildewideδ > δ. Then a subsequence of Gαconverges at least in the Cq−1(Ha) topology to a map G∞ofHainto/tildewideHwhich is a harmonic map from Hainto/tildewideHand takes∂Hato a constant mean curvature hypersurface G∞(∂Ha) of (/tildewideH,/tildewidehij) with the areaa, as well as satisﬁes the free boundary condition. Clearly, G∞is at least a local diﬀeomorphism. Since G∞is the limit of diﬀeomorphisms, the only possibility of overlap is at the boundary. Note that G∞(∂Ha) is still strictly concave. So G∞ is still a diﬀeomorphism. Moreover by using the standard reg ularity result of elliptic partial diﬀerential equations (see for example [48]), we al so have (7.6.20) /ba∇dblG∗ ∞/tildewidehij−hij/ba∇dblCq(Ha)=δ. Now by Lemma 5.3.7 we deduce that there exists an isometry IofHto/tildewideHwith dCl0(Ha)(G∞,I)<ǫ0. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 479 ThusI−1◦G∞is a harmonic diﬀeomorphism of Hato itself which satisﬁes the free boundary condition and dCl0(Ha)(I−1◦G∞,Id)<ǫ0. However the uniqueness in Lemma 5.3.8 concludes that I−1◦G∞=Idwhich contra- dicts (7.6.20). So we have shown that at least one hyperbolic piece persists and the metrict−1F∗ α(t)gij(t), forωα≤t<∞, is as close to the hyperbolic metric hijas we like. We can continue to form other persistent hyperbolic pieces i n the same way as long as there is a sequence of points yβ∈Mthick(w,tβ),tβ→+∞, lying outside the chosen pieces. Note that V(t)(t+3 2)−3 2is nonincreasing on [0 ,+∞). Therefore by combining with Margulis lemma (see for example [55] or [76]), we have pr oved that there exists a ﬁnite collection of complete noncompact hyperbolic three- manifolds H1,...,Hmwith ﬁnite volume, a small number a >0 and a time T <+∞such that for all tbeyond Twe can ﬁnd diﬀeomorphisms ϕl(t) of (Hl)aintoMt, 1≤l≤m, so that as long as t is suﬃciently large, the metric t−1ϕ∗ l(t)gij(t) is as close to the hyperbolic metrics as we like and the complement Mt\(ϕ1(t)((H1)a)∪ ··· ∪ϕm(t)((Hm)a)) is contained in the thin part Mthin(w,t). It remains to show the boundary tori of any persistent hyperb olic piece are incom- pressible. Let Bbe a small positive number and assume the above positive numb er ais much smaller than B. LetMa(t) =ϕl(t)((Hl)a) (1≤l≤m) be such a per- sistent hyperbolic piece of the manifold Mttruncated by boundary tori of area at with constant mean curvature, and denote by Mc a(t) =Mt\◦ Ma(t) the part of Mt exterior to Ma(t). Thus there exists a family of subsets MB(t)⊂Ma(t) which is a persistent hyperbolic piece of the manifold Mttruncated by boundary tori of area Bt with constant mean curvature. We also denote by Mc B(t) =Mt\◦ MB(t). By Van Kampen’s Theorem, if π1(∂MB(t)) injects into π1(Mc B(t)) then it injects into π1(Mt) also. Thus we only need to show π1(∂MB(t)) injects into π1(Mc B(t)). As before we will use a contradiction argument to show π1(∂MB(t)) injects into π1(Mc B(t)). LetTbe a torus in ∂MB(t) and suppose π1(T) does not inject into π1(Mc B(t)). By Dehn’s Lemma we know that the kernel is a cyclic subgrou p ofπ1(T) generated by a primitive element. Consider the normalized m etric/tildewidegij(t) =t−1gij(t) onMt. Then by the work of Meeks-Yau [86] or Meeks-Simon-Yau [87], we know that among all disks in Mc B(t) whose boundary curve lies in Tand generates the kernel ofπ1(T), there is a smooth embedded disk normal to the boundary whic h has the least possible area (with respect to the normalized metric /tildewidegij(t)). Denote by Dthe minimal disk and /tildewideA=/tildewideA(t) its area. We will show that /tildewideA(t) decreases at a certain rate which will arrive at a contradiction. We ﬁrst consider the case that there exist no surgeries at the timet. Exactly as in Part III of the proof of Theorem 5.3.4, the change of the are a/tildewideA(t) comes from the change in the metric and the change in the boundary. For the ch ange in the metric, we choose an orthonormal frame X,Y,Z at a pointxin the disk Dso thatXandY are tangent to the disk DwhileZis normal. Since the normalized metric /tildewidegijevolves by ∂ ∂t/tildewidegij=−t−1(/tildewidegij+ 2/tildewideRij), 480 H.-D. CAO AND X.-P. ZHU the (normalized) area element d/tildewideσof the disk Daroundxsatisﬁes ∂ ∂td/tildewideσ=−t−1(1 +/tildewidestRic (X,X) +/tildewidestRic (Y,Y))d/tildewideσ. For the change in the boundary, we notice that the tensor /tildewidegij+ 2/tildewideRijis very small for the persistent hyperbolic piece. Then by using the Gauss-Bo nnet theorem as before, we obtain the rate of change of the area (7.6.21)d/tildewideA dt≤ −/integraldisplay D/parenleftigg 1 t+/tildewideR 2t/parenrightigg d/tildewideσ+1 t/integraldisplay ∂D/tildewidekd/tildewides−2π t+o/parenleftbigg1 t/parenrightbigg /tildewideL, where/tildewidekis the geodesic curvature of the boundary and /tildewideLis the length of the boundary curve∂D(with respect to the normalized metric /tildewidegij(t)). Since/tildewideR≥ −3t/2(t+3 2) for allt≥0 by (7.6.2), the ﬁrst term on the RHS of (7.6.21) is bounded ab ove by −/integraldisplay D/parenleftigg 1 t+/tildewideR 2t/parenrightigg d/tildewideσ≤ −1 t/parenleftbigg1 4−o(1)/parenrightbigg /tildewideA; while the second term on the RHS of (7.6.21) can be estimated e xactly as before by 1 t/integraldisplay ∂D/tildewidekd/tildewides≤1 t/parenleftbigg1 4+o(1)/parenrightbigg /tildewideL. Thus we obtain (7.6.22)d/tildewideA dt≤1 t/bracketleftbigg/parenleftbigg1 4+o(1)/parenrightbigg /tildewideL−/parenleftbigg1 4−o(1)/parenrightbigg /tildewideA−2π/bracketrightbigg . Next we show that these arguments also work for the case that t here exist surgeries at the time t. To this end, we only need to check that the embedded minimal d isk Dlies in the region which is unaﬀected by surgery. Our surgeri es for the irreducible three-manifold took place on δ-necks inε-horns, where the scalar curvatures are at leastδ−2(/tildewider(t))−1, and the components with nonnegative scalar curvature have been removed. So the hyperbolic piece is not aﬀected by the surger ies. In particular, the boundary∂Dis unaﬀected by the surgeries. Thus if surgeries occur on the minimal disk, the minimal disk has to pass through a long thin neck bef ore it reaches the surgery regions. Look at the intersections of the embedding minimal disk with a generic center two-sphere S2of the long thin neck; these are circles. Since the two- sphere S2is simply connected, we can replace the components of the min imal disk Doutside the center two-sphere S2by some corresponding components on the center two-sphere S2to form a new disk which also has ∂Das its boundary. Since the metric on the long thin neck is nearly a product metric, we could choo se the generic center two-sphere S2properly so that the area of the new disk is strictly less than the area of the original disk D. This contradiction proves the minimal disk lies entirely i n the region unaﬀected by surgery. Sinceais much smaller than B, the region within a long distance from ∂MB(t) intoMc B(t) will look nearly like a hyperbolic cusplike collar and is un aﬀected by the surgeries. So we can repeat the arguments in the last part of t he proof of Theorem 5.3.4 to bound the length /tildewideLby the area/tildewideAand to conclude d/tildewideA dt≤ −π t THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 481 for all suﬃciently large times t, which is impossible because the RHS is not inte- grable. This proves that the boundary tori of any persistent hyperbolic piece are incompressible. Therefore we have proved the theorem. 7.7. Geometrization of Three-manifolds. In the late 70’s and early 80’s, Thurston [122], [123] [124] proved a number of remarkable re sults on the existence of geometric structures on a class of three-manifolds: Haken manifolds (i.e. each of them contains an incompressible surface of genus ≥1). These results motivated him to formulate a profound conjecture which roughly says every compact three-manifold admits a canonical decomposition into domains, each of whic h has a canonical geo- metric structure. To give a detailed description of the conj ecture, we recall some terminology as follows. Ann-dimensional complete Riemannian manifold ( M,g) is called a homoge- neous manifold if its group of isometries acts transitively on the manifold . This means that the homogeneous manifold looks the same metrical ly at everypoint. For example, the round n-sphere Sn, the Euclidean space Rnand the standard hyperbolic space Hnare homogeneous manifolds. A Riemannian manifold is said to be mod- eledon a given homogeneous manifold ( M,g) if every point of the manifold has a neighborhood isometric to an open set of ( M,g). And ann-dimensional Riemannian manifold is called a locally homogeneous manifold if it is complete and is modeled on a homogeneous manifold. By a theorem of Singer [119], the u niversal cover of a locally homogeneous manifold (with the pull-back metric) i s a homogeneous manifold. In dimension three, every locally homogeneous manifold wit h ﬁnite volume is modeled on one of the following eight homogeneous manifolds (see for example The- orem 3.8.4 of [125]): (1)S3, the round three-sphere; (2)R3, the Euclidean space ; (3)H3, the standard hyperbolic space; (4)S2×R; (5)H2×R; (6)Nil, the three-dimensional nilpotent Heisenberg group (consi sting of upper triangular 3 ×3 matrices with diagonal entries 1); (7)/tildewidePSL(2,R), the universal cover of the unit sphere bundle of H2; (8)Sol, the three-dimensional solvable Lie group. A three-manifold Mis called prime if it is not diﬀeomorphic to S3and if every (topological) S2⊂M, which separates Minto two pieces, has the property that one of the two pieces is diﬀeomorphic to a three-ball. Recall tha t a three-manifold is irreducible if every embedded two-sphere bounds a three-ba ll in the manifold. Clearly an irreducible three-manifold is either prime or is diﬀeomo rphic to S3. Conversely, an orientable prime three-manifold is either irreducible or i s diﬀeomorphic to S2×S1(see for example [69]). One of the ﬁrst results in three-manifold topology is the following prime decomposition theorem obtained by Kneser [79] in 1929 (see also Theorem 3.15 of [69]). Prime Decomposition Theorem. Every compact orientable three-manifold admits a decomposition as a ﬁnite connected sum of orientabl e prime three-manifolds. In [90], Milnor showed that the factors involved in the above Prime Decompo- sition are unique. Based on the prime decomposition, the que stion about topology 482 H.-D. CAO AND X.-P. ZHU of compact orientable three-manifolds is reduced to the que stion about prime three- manifolds. Thurston’s Geometrization Conjecture is about prime three-manifolds. Thurston’s Geometrization Conjecture. LetMbe a compact, orientable and prime three-manifold. Then there is an embedding of a ﬁni te number of disjoint unions, possibly empty, of incompressible two-tori/coproducttext iT2 i⊂Msuch that every com- ponent of the complement admits a locally homogeneous Riema nnian metric of ﬁnite volume. We remark that the existence of a torus decomposition, also c alled JSJ- decomposition, was already obtained by Jaco-Shalen [74] an d Johannsen [75]. The JSJ-decomposition states that any compact, orientable, an d prime three-manifold has a ﬁnite collection, possibly empty, of disjoint incompress ible embedding two-tori {T2 i} which separate the manifold into a ﬁnite collection of compa ct three-manifolds (with toral boundary), each of which is either a graph manifold or i satoroidal in the sense that any subgroup of its fundamental group isomorphic to Z×Zis conjugate into the fundamental group of some component of its boundary. A compa ct three-manifold X, possibly with boundary, is called a graph manifold if there is a ﬁnite collection of disjoint embedded tori Ti⊂Xsuch that each component of X\/uniontextTiis an S1bundle over a surface. Thus the point of the conjecture is that the co mponents should all be geometric. The geometrization conjecture for a general compact orient able 3-manifold is the statement that each of its prime factors satisﬁes the abo ve conjecture. We say a compact orientable three-manifold is geometrizable if it satisﬁes the geometric conjecture. We also remark that the Poincar´ e conjecture can be deduced f rom Thurston’s geometrization conjecture. Indeed, suppose that we have a c ompact simply connected three-manifold that satisﬁes the conclusion of the geometr ization conjecture. If it were not diﬀeomorphic to the three-sphere S3, there would be a prime factor in the prime decomposition of the manifold. Since the prime factor still has vanishing fundamental group, the (torus) decomposition of the prime factor in the g eometrization conjecture must be trivial. Thus the prime factor is a compact homogeneo us manifold model. From the list of above eight models, we see that the only compa ct three-dimensional model is S3. This is a contradiction. Consequently, the compact simply connected three-manifold is diﬀeomorphic to S3. Now we apply the Ricci ﬂow to discuss Thurston’s geometrizat ion conjecture. LetMbe a compact, orientable and prime three-manifold. Since a p rime orientable three-manifold is either irreducible or is diﬀeomorphic to S2×S1, we may thus assume the manifold Mis irreducible also. Arbitrarily given a (normalized) Riem annian metric for the manifold M, we use it as initial data to evolve the metric by the Ricci ﬂow with surgery. From Theorem 7.4.3, we know that the R icci ﬂow with surgery has a long-time solution on a maximal time interval [ 0,T) which satisﬁes the a priori assumptions and has a ﬁnite number of surgeries on ea ch ﬁnite time interval. Furthermore, from the long-time behavior theorem (Theorem 7.6.4), we have well- understood geometric structures on the thick part. Whereas , to understand the thin part, Perelman announced the following assertion in [104]. Perelman’s Claim ([104]).Suppose (Mα,gα ij) is a sequence of compact ori- entable three-manifolds, closed or with convex boundary, a ndwα→0. Assume that (1) for each point x∈Mαthere exists a radius ρ=ρα(x),0< ρ < 1,not exceeding the diameter of the manifold, such that the ball B(x,ρ) in the THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 483 metricgα ijhas volume at most wαρ3and sectional curvatures at least −ρ−2; (2) each component of the boundary of Mαhas diameter at most wα, and has a (topologically trivial) collar of length one, where the sec tional curvatures are between −1/4−ǫand−1/4 +ǫ. ThenMαfor suﬃciently large αare diﬀeomorphic to graph manifolds. The topology of graph manifolds is well understood; in parti cular, every graph manifold is geometrizable (see [126]). The proof of Perelman’s Claim promised in [104] is still not a vailable in literature. Nevertheless, recently in [118], Shioya and Yamaguchi prov ided a proof of Perelman’s Claim for the special case that all the manifolds ( Mα,gα ij) are closed. That is, they proposed a proof for the following weaker assertion. Weaker Assertion (Theorem 8.1 of Shioya-Yamaguchi [118]) .Suppose (Mα,gα ij) is a sequence of compact orientable three-manifolds witho ut boundary, and wα→0. Assume that for each point x∈Mαthere exists a radius ρ=ρα(x), not exceeding the diameter of the manifold, such that the bal lB(x,ρ) in the metric gα ijhas volume at most wαρ3and sectional curvatures at least −ρ−2. ThenMαfor suﬃciently large αare diﬀeomorphic to graph manifolds. Based on the the long-time behavior theorem (Theorem 7.6.4) and assuming the above Weaker Assertion, we can now give a proof for Thurston’ s geometrization con- jecture . We remark that if we assume the above Perelman’s Cla im, then we does not need to use Thurston’s theorem for Haken manifolds in the pro of of Theorem 7.7.1. Theorem 7.7.1. Thurston’s geometrization conjecture is true. Proof. LetMbe a compact, orientable, and prime three-manifold (withou t boundary). Without loss of generality, we may assume that th e manifold Mis ir- reducible also. Recall that the theorem of Thurston (see for example Theorem A and Theorem B in Section 3 of [94], see also [85] and [102]) says that any co mpact, orientable, and irreducible Haken three-manifold (with or without boun dary) is geometrizable. Thus in the following, we may assume that the compact three-m anifoldM(without boundary) is atoroidal, and then the fundamental group π1(M) contains no noncyclic, abelian subgroup. Arbitrarily given a (normalized) Riemannian metric on the m anifoldM, we use it as initial data for the Ricci ﬂow. Arbitrarily take a seque nce of small positive constantswα→0 asα→+∞. For each ﬁxed α, we setε=wα/2>0. Then by Theorem 7.4.3, the Ricci ﬂow with surgery has a long-time sol ution (Mα t,gα ij(t)) on a maximal time interval [0 ,Tα), which satisﬁes the a priori assumptions (with the accuracy parameter ε=wα/2) and has a ﬁnite number of surgeries on each ﬁnite time interval. Since the initial manifold is irreducible, b y the surgery procedure, we know that for each αand eacht >0 the solution manifold Mα tconsists of a ﬁnite number of components where the essential component ( Mα t)(1)is diﬀeomorphic to the initial manifold Mand the others are diﬀeomorphic to the three-sphere S3. If for someα=α0the maximal time Tα0is ﬁnite, then the solution ( Mα0 t,gα0 ij(t)) becomes extinct at Tα0and the (irreducible) initial manifold Mis diﬀeomorphic to S3/Γ (the metric quotients of round three-sphere); in particul ar, the manifold Mis geometrizable. Thus we may assume that the maximal time Tα= +∞for allα. We now apply the long-time behavior theorem (Theorem 7.6.4) . If there is some αsuch that case (ii) of Theorem 7.6.4 occurs, then for some suﬃ ciently large time 484 H.-D. CAO AND X.-P. ZHU t, the essential component ( Mα t)(1)of the solution manifold Mα tis diﬀeomorphic to a compact hyperbolic space, so the initial manifold Mis geometrizable. Whereas if there is some suﬃciently large αsuch that case (iii) of Theorem 7.6.4 occurs, then it follows that for all suﬃciently large t, there is an embedding of a (nonempty) ﬁnite number of disjoint unions of incompressible two-tori/coproducttext iT2 iin the essential component (Mα t)(1)ofMα t. This is a contradiction since we have assumed the initial ma nifold Mis atoroidal. It remains to deal with the situation that there is a sequence of positive αk→ +∞such that the solutions ( Mαk t,gαk ij(t)) always satisfy case (i) of Theorem 7.6.4. That is, for each αk,Mαk t=Mthin(wαk,t) when the time tis suﬃciently large. By the Thick-thin decomposition theorem (Theorem 7.6.3), the re is a positive constant, 0< ρ(wαk)≤1, such that as long as tis suﬃciently large, for every x∈Mαk t= Mthin(wαk,t), we have some r=r(x,t), with 0<r√ t<ρ(wαk)√ t, such that (7.7.1) Rm≥ −(r√ t)−2onBt(x,r√ t), and (7.7.2) Vol t(Bt(x,r√ t))<wαk(r√ t)3. Clearly we only need to consider the essential component ( Mαk t)(1). We divide the discussion into the following two cases: (1) there is a positive constant 1 < C < +∞such that for each αkthere is a suﬃciently large time tk>0 such that (7.7.3) r(x,tk)√tk<C·diam/parenleftig (Mαk tk)(1)/parenrightig for allx∈(Mαk tk)(1)⊂Mthin(wαk,tk); (2) there are a subsequence αk(still denoted by αk), and sequences of positive constantsCk→+∞and timesTk<+∞such that for each t≥Tk, we have (7.7.4) r(x(t),t)√ t≥Ck·diam/parenleftig (Mαk t)(1)/parenrightig for somex(t)∈(Mαk t)(1),k= 1,2,....Here we denote by diam(( Mα t)(1)) the diam- eter of the essential component ( Mα t)(1)with the metric gα ij(t) at the time t. Let us ﬁrst consider case (1). For each point x∈(Mαk tk)(1)⊂Mthin(wαk, tk), we denote byρk(x) =C−1r(x,tk)√tk. Then by (7.7.1), (7.7.2) and (7.7.3), we have ρk(x)<diam/parenleftig (Mαk tk)(1)/parenrightig , Voltk(Btk(x,ρk(x)))≤Voltk(Btk(x,r(x,tk)√tk))<C3wαk(ρk(x))3, and Rm≥ −(r(x,tk)√tk)−2≥ −(ρk(x))−2 onBtk(x,ρk(x)).Then it follows from the above Weaker Assertion that ( Mαk tk)(1), for suﬃciently large k, are diﬀeomorphic to graph manifolds. This implies that the (irreducible) initial manifold Mis diﬀeomorphic to a graph manifold. So the manifold Mis geometrizable in case (1). THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 485 We next consider case (2). Clearly, for each αkand the chosen Tk, we may assume that the estimates (7.7.1) and (7.7.2) hold for all t≥Tkandx∈(Mαk t)(1). The combination of (7.7.1) and (7.7.4) gives (7.7.5) Rm≥ −C−2 k(diam((Mαk t)(1)))−2on (Mαk t)(1), for allt≥Tk. If there are a subsequence αk(still denoted by αk) and a sequence of timestk∈(Tk,+∞) such that (7.7.6) Vol tk((Mαk tk)(1))<w′ k(diam((Mαk tk)(1)))3 for some sequence w′ k→0, then it follows from the Weaker Assertion that ( Mαk tk)(1), for suﬃciently large k, are diﬀeomorphic to graph manifolds which implies the init ial manifoldMis geometrizable. Thus we may assume that there is a positive constant w′such that (7.7.7) Vol t((Mαk t)(1))≥w′(diam((Mαk t)(1)))3 for eachkand allt≥Tk. In view of the estimates (7.7.5) and (7.7.7), we now want to us e Theorem 7.5.2 to get a uniform upper bound for the curvatures of the es sential components ((Mαk t)(1),gαk ij(t)) with suﬃciently large time t. Note that the estimate in Theorem 7.5.2 depends on the parameter εand ourε’s depend on wαkwith 0<ε=wαk/2; so it does not work in the present situation. Fortunately we not ice that the curvature estimate for smooth solutions in Corollary 7.2.3 is indepen dent ofε. In the following we try to use Corollary 7.2.3 to obtain the desired curvature estimate. We ﬁrst claim that for each k, there is a suﬃciently large T′ k∈(Tk,+∞) such that the solution, when restricted to the essential compone nt ((Mαk t)(1),gαk ij(t)), has no surgery for all t≥T′ k. Indeed, for each ﬁxed k, if there is a δ(t)-cutoﬀ surgery at a suﬃciently large time t, then the manifold (( Mαk t)(1),gαk ij(t)) would contain a δ(t)-neckBt(y,δ(t)−1R(y,t)−1 2) for somey∈(Mαk t)(1)with the volume ratio (7.7.8)Volt(Bt(y,δ(t)−1R(y,t)−1 2)) (δ(t)−1R(y,t)−1 2)3≤8πδ(t)2. On the other hand, by (7.7.5) and (7.7.7), the standard Bisho p-Gromov volume com- parison implies that Volt(Bt(y,δ(t)−1R(y,t)−1 2)) (δ(t)−1R(y,t)−1 2)3≥c(w′) for some positive constant c(w′) depending only on w′. Sinceδ(t) is very small when tis large, this arrives at a contradiction with (7.7.8). So fo r eachk, the essential component (( Mαk t)(1),gαk ij(t)) has no surgery for all suﬃciently large t. For eachk, we consider any ﬁxed time ˜tk>3T′ k. Let us scale the solution gαk ij(t) on the essential component ( Mαk t)(1)by ˜gαk ij(·,s) = (˜tk)−1gαk ij(·,˜tks). Note that ( Mαk t)(1)is diﬀeomorphic to Mfor allt. By the above claim, we see that the rescaled solution ( M,˜gαk ij(·,s)) is a smooth solution to the Ricci ﬂow on the time intervals∈[1 2,1]. Set ˜rk=/parenleftig/radicalbig ˜tk/parenrightig−1 diam/parenleftig (Mαk ˜tk)(1)/parenrightig . 486 H.-D. CAO AND X.-P. ZHU Then by (7.7.4), (7.7.5) and (7.7.7), we have ˜rk≤C−1 k→0,ask→+∞, /tildewidestRm≥ −C−2 k(˜rk)−2,onB1(x(˜tk),˜rk), and Vol1(B1(x(˜tk),˜rk))≥w′(˜rk)3, where/tildewidestRmis the rescaled curvature, x(˜tk) is the point given by (7.7.4) and B1(x(˜tk),˜rk) is the geodesic ball of rescaled solution at the time s= 1. Moreover, the closure ofB1(x(˜tk),˜rk) is the whole manifold ( M,˜gαk ij(·,1)). Note that in Theorem 7.2.1, Theorem 7.2.2 and Corollary 7.2. 3, the condition about normalized initial metrics is just to ensure that the s olutions satisfy the Hamilton-Ivey pinching estimate. Since our solutions ( Mαk t,gαk ij(t)) have already satisﬁed the pinching assumption, we can then apply Corolla ry 7.2.3 to conclude |/tildewidestRm(x,s)| ≤K(w′)(˜rk)−2, whenevers∈[1−τ(w′)(˜rk)2,1],x∈(M,˜gαk ij(·,s)) andkis suﬃciently large. Here K(w′) andτ(w′) are positive constants depending only on w′. Equivalently, we have the curvature estimates (7.7.9) |Rm(·,t)| ≤K(w′)(diam((Mαk ˜tk)(1)))−2,onM, whenevert∈[˜tk−τ(w′)(diam ((Mαk ˜tk)(1)))2,˜tk] andkis suﬃciently large. For eachk, let us scale (( Mαk t)(1),gαk ij(t)) with the factor (diam(( Mαk ˜tk)(1)))−2and shift the time ˜tkto the new time zero. By the curvature estimate (7.7.9) and Ha milton’s compactness theorem (Theorem 4.1.5), we can take a subseque ntial limit (in the C∞ topology) and get a smooth solution to the Ricci ﬂow on M×(−τ(w′),0]. Moreover, by (7.7.5), the limit has nonnegative sectional curvature o nM×(−τ(w′),0]. Recall that we have removed all compact components with nonnegativ e scalar curvature. By combining this with the strong maximum principle, we conclu de that the limit is a ﬂat metric. Hence in case (2), Mis diﬀeomorphic to a ﬂat manifold and then it is also geometrizable. Therefore we have completed the proof of the theorem. REFERENCES [1]U. Abresch and W.T. Meyer ,Injectivity radius estimates and sphere theorems, Comparison Geometry, MSRI Publications, 30 (1997), pp. 1–48. [2]S. Aloff and N. Wallach ,An inﬁnite family of distinct 7-manifolds admitting positively curved Riemannian structure , Bull. Amer. Math. Soc., 81 (1975), pp. 93–97. [3]W. Ambrose and I.M. Singer ,A theorem on holonomy, Trans. Amer. Math. Soc., 75 (1953), pp. 428–443. [4]M. Anderson ,Geometrization of 3-manifolds via the Ricci ﬂow , Notices Amer. Math. Soc., 51:2 (2004), pp. 184–193. [5]S. Bando ,On the classiﬁcation of three-dimensional compact K¨ ahler manifolds of nonnega- tive bisectional curvature, J. Diﬀerential Geom., 19:2 (1984), pp. 283–297. [6]J. Bartz, M. Struwe, and R. Ye ,A new approach to the Ricci ﬂow on S2, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21:3 (1994), pp. 475–482. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 487 [7]M. Berger ,Sur les groupes d’holonomie des vari´ et´ es ` a connexion aﬃn e et les vari´ et´ es rie- manniennes , Bull. Soc. Math. France, 83 (1955), pp. 279–330. [8]Y. Burago, M. Gromov, and G. Perelman ,A.D. Alexandrov spaces with curvatures bounded below , Russian Math. Surveys, 47 (1992), pp. 1–58. [9]D. Burago, Y. Burago, and S. Ivanov ,A Course in Metric Geometry, Graduate Studies in Mathematics, 33, Amer. Math. Soc. Providence, Rhode Isla nd. [10]R. Bryant ,Local existence of gradient Ricci solitons , preprint. [11]H.-D. Cao ,Deformation of K¨ ahler metrics to K¨ ahler-Einstein metric s on compact K¨ ahler manifolds , Invent. Math., 81:2 (1985), pp. 359–372. [12] ,On Harnack’s inequalities for the K ¨ahler-Ricci ﬂow, Invent. Math., 109:2 (1992), pp. 247–263. [13] ,Existence of gradient K¨ ahler-Ricci solitons, Elliptic an d Parabolic Methods in Geom- etry(Minneapolis, MN, 1994), pp. 1–16, A K Peters, Wellesley, MA , 1996. [14] ,Limits of Solutions to the K¨ ahler-Ricci ﬂow , J. Diﬀerential Geom., 45 (1997), pp. 257– 272. [15]H.-D. Cao, B.L. Chen, and X.P. Zhu ,Ricci ﬂow on compact K¨ ahler manifolds of positive bisectional curvature, C.R. Math. Acad. Sci. Paris, 337:12 (2003), pp. 781–784. [16]H.-D. Cao and B. Chow ,Recent developments on the Ricci ﬂow , Bull. Amer. Math. Soc (N.S.), 36:1 (1999), pp. 59–74. [17]H.-D. Cao, B. Chow, S.C. Chu, and S.-T. Yau ,Collected Papers on Ricci Flow , Series in Geometry and Topology, 37, International Press, 2003. [18]H.-D. Cao, R.S. Hamilton, and T. Ilmanen ,Gaussian densities and stability for some Ricci solitons, arXiv:DG.math.DG/0404165. [19]J. Cheeger ,Finiteness theorems for Riemannian manifolds , Amer. J. Math., 92 (1970), pp. 61–74. [20]J. Cheeger and T.H. Colding ,Lower bounds on Ricci curvature and the almost rigidity of warped products , Ann. of Math. (2), 144:1 (1996), pp. 189–237. [21] ,On the structure of the spaces with Ricci curvature bounded b elow, I, J. Diﬀerential Geom., 46 (1997), pp. 406–480. [22]J. Cheeger and D. Ebin ,Comparison theorems in Riemannian geometry, North-Holland, 1975. [23]J. Cheeger and D. Gromoll ,On the structure of complete manifolds of nonnegative cur- vature, Ann. of Math., 46 (1972), pp. 413–433. [24]J. Cheeger and M. Gromov ,Collapsing Riemannian manifolds while keeping their curva ture bounded , I, J. Diﬀerential Geom., 23:3 (1986), pp. 309–346. [25] ,Collapsing Riemannian manifolds while keeping their curva ture bounded , II, J. Dif- ferential Geom., 32:1 (1990), pp. 269–298. [26]J. Cheeger, M. Gromov, and K. Fukaya ,Mikhael Nilpotent structures and invariant met- rics on collapsed manifolds, J. Amer. Math. Soc., 5:2 (1992), pp. 327–372. [27]J. Cheeger, M. Gromov, and M. Taylor ,Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of compl ete Riemannian manifolds, J. Diﬀerential Geom., 17 (1982), pp. 15–53. [28]J. Cheeger and S.-T. Yau ,A lower bound for the heat kernel , Comm. on Pure and Appl. Math., 34 (1981), pp. 465–480. [29]B.L. Chen, S.H. Tang, and X.P. Zhu ,A uniformization theorem of complete noncompact K¨ ahler surfaces with positive bisectional curvature , J. Diﬀerential Geometry, 67 (2004), pp. 519–570. [30]B.L. Chen and X.P. Zhu ,Complete Riemannian manifolds with pointwise pinched curv ature, Invent. Math., 140:2 (2000), pp. 423–452. [31] ,On complete noncompact K¨ ahler manifolds with positive bis ectional curvature , Math. Ann., 327 (2003), pp. 1–23. [32] ,Volume growth and curvature decay of positively curved K¨ ah ler manifolds , Quarterly Journal of Pure and Applied Mathematics, 1:1 (2005), pp. 68– 108. [33] ,Uniqueness of the Ricci Flow on Complete Noncompact Manifol ds,arXiv:DG (AP) math. DG/0505447, May 2005. [34] ,Ricci Flow with Surgery on Four-manifolds with Positive Iso tropic Curvature, arXiv:math.DG/0504478, April 2005. [35]S.Y. Cheng, P. Li, and S.-T. Yau ,On the upper estimate of the heat kernel of complete Riemannian manifold, Amer. J. Math., 103:5 (1981), pp. 1021–1063. [36]B. Chow ,The Ricci ﬂow on 2-sphere , J. Diﬀerential Geom., 33 (1991), pp. 325–334. [37] ,On the entropic estimate for the Ricci ﬂow on compact 2-orbifolds , J. Diﬀerential Geom., 33 (1991), pp. 597–600. 488 H.-D. CAO AND X.-P. ZHU [38]B. Chow and S.C. Chu ,A geometric interpretation of Hamilton’s Harnack inequali ty for the Ricci ﬂow, Math. Res. Lett., 2 (1995), pp. 701–718. [39]B. Chow and D. Knopf ,The Ricci ﬂow: An introduction , Mathematical Surveys and Mono- graphs, Amer. Math. Soc., Providence, RI, 2004. [40]B. Chow and P. Lu ,The maximum principle for systems of parabolic equations su bject to an avoidance set, Paciﬁc J. Math., 214:2 (2004), pp. 201–222. [41]B. Chow, P. Lu, and L. Ni ,Hamilton’s Ricci ﬂow , American Mathematical Society and Science Press, to appear. [42]T.H. Colding and W.P. Minicozzi, II ,Estimates for the extinction time for the Ricci ﬂow on certain 3-manifolds and a question of Perelman, J. Amer. Math. Soc., 18:3 (2005), pp. 561–569. [43]D. De Turck ,Deforming metrics in the direction of their Ricci tensors , J. Diﬀerential Geom., 18 (1983), pp. 157–162. [44]G. Drees ,Asymptotically ﬂat manifold of nonnegative curvature , Diﬀ. Geom. Appl., 4 (1994), pp. 77–90. [45]J. Eschenburg, V. Schroeder, and M. Strake ,Curvature at inﬁnity of open nonnegatively curved manifold , J. Diﬀerential Geom., 30 (1989), pp. 155–166. [46]M. Feldman, T. Ilmanen, and D. Knopf ,Rotationally symmetric shrinking and expanding K¨ ahler-Ricci solitons , J. Diﬀerential Geom., 65 (2003), pp. 169–209. [47]T. Frankel ,Manifolds with positive curvature , Paciﬁc J. Math., 11 (1961), pp. 165–174. [48]D. Gilbarg and N.S. Trudinger ,Elliptic partial diﬀerential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Sprin ger-Verlag, Berlin, 2001. [49]R.E. Greene and H. Wu ,On a new gap phenomonon in Riemannian geometry , Proc. Nat. Acad. Sci. U.S.A., 79:2 (1982), pp. 714–715. [50] ,Gap theorems for noncompact Riemannian manifolds , Duke Math. J., 49 (1982), pp. 731–756. [51] ,Lipschitz convergence of Riemannian manifolds , Paciﬁc J. Math., 131 (1988), pp. 119–141. [52]D. Gromoll and W. Meyer ,On complete open manifolds of positive curvature , Ann. of Math., 90 (1969), pp. 75–90. [53]M. Gromov ,Metric Structures for Riemannian and Non-Riemannian Space s, Edited by J. LaFontaine and P. Pansu, English translation by Sean Mich ael Bates, Birkh¨ auser, 1998. [54] ,Almost ﬂat manifolds , J. Diﬀerential Geom., 13 (1978), pp. 231–241. [55] ,Hyperbolic manifolds (according to Thurston and J φrgensen), Bourbaki Seminar, Vol. 1979/80, pp. 40–53, Lecture Notes in Math., 842, Springer, B erlin-New York, 1981. [56]M. Gromov and H.B. Lawson, Jr. ,The classiﬁcation of simply connected manifolds of positive scalar curvature, Ann. of Math. (2), 111:3 (1980), pp. 423–434. [57] ,Positive scalar curvature and the Dirac operator on complet e Riemannian manifolds, Inst. Hautes ´Etudes Sci. Publ. Math., 58 (1983), pp. 83–196. [58]R.S. Hamilton ,Three manifolds with positive Ricci curvature , J. Diﬀerential Geom., 17 (1982), pp. 255–306. [59] ,Four–manifolds with positive curvature operator , J. Diﬀerential Geom., 24 (1986), pp. 153–179. [60] ,The Ricci ﬂow on surfaces , Contemporary Mathematics, 71 (1988), pp. 237–261. [61] ,The Harnack estimate for the Ricci ﬂow , J. Diﬀerential Geom., 37 (1993), pp. 225- 243. [62] ,A compactness property for solution of the Ricci ﬂow , Amer. J. Math., 117 (1995), pp. 545–572. [63] ,The formation of singularities in the Ricci ﬂow , Surveys in Diﬀerential Geometry, 2, pp. 7–136, International Press, 1995. [64] ,Four manifolds with positive isotropic curvature , Comm. in Analysis and Geometry, 5 (1997), pp. 1–92. [65] ,Non-singular solutions to the Ricci ﬂow on three manifolds , Comm. Anal. Geom., 7 (1999), pp. 695–729. [66] ,Eternal solutions to the Ricci ﬂow , J. Diﬀerential Geom., 38:1 (1993), pp. 1–11. [67] ,Lectures on the Ricci ﬂow at Columbia University , Fall, 2003. [68]S. Helgason ,Diﬀerential Geometry and Symmetric Spaces , Academic Press, 1962 (Second edition, 1978). [69]J. Hempel ,3-manifolds, Reprint of the 1976 original, AMS Chelsea Publishing, Provi dence, RI, 2004. [70]M.W. Hirsch ,Diﬀerential Topology , Springer-Verlag, 1976. THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 489 [71]G. Huisken ,Ricci deformation of the metric on a Riemanian manifold, J. Diﬀerential Geom- etry, 21 (1985), pp. 47–62. [72] ,Asymptotic behavior for singularities of the mean curvatur e ﬂow. J. Diﬀerential Geometry, 31 (1990), pp. 285–299. [73]T. Ivey ,Ricci solitons on compact three-manifolds , Diﬀ. Geom. Appl., 3 (1993), pp. 301–307. [74]W. Jaco and P. Shalen ,Seifert ﬁbered spaces in 3-manifolds , Men. Amer. Math. Soc., 220 (1979). [75]K. Johannson ,Homotopy equivalences of 3-manifolds with boundaries , Lecture Notes in Math., 761 (1979). [76]D.A. Kazdan and G. Margulis ,A simple proof of Selberg’s hypothesis, Math. Sb. (117), 75 (1968), pp. 163–168. [77]J.L. Kazdan and F.W. Warner ,Curvature functions for compact 2-manifolds , Ann. of Math. (2), 99 (1974), pp. 14–47. [78]B. Kleiner and J. Lott ,Note on Perelman’s paper , http://www.math.lsa.umich.edu /re- search /ricciﬂow /perelman.html. [79]H. Kneser ,Geschlossene Fl¨ achen in dreidmensionalen mannigfaltigk eiten, Jahresbericht der Deut. Math. Verein., 38 (1929), pp. 248–260. [80]N. Koiso ,On rotationally symmmetric Hamilton’s equation for K¨ ahle r-Einstein metrics , Recent Topics in Diﬀ. Anal. Geom. Adv. Studies Pure Math., 18 -I (1990), pp. 327–337. [81]O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Uralceva ,Linear and quasilinear equa- tions of parabolic type, Transl. Amer. Math. Soc., 23 (1968). [82]P. Li and S.-T. Yau ,On the parabolic kernel of the Schr¨ odinger operator, Acta Math., 156:3-4 (1986), pp. 153–201. [83]C. Margerin ,Une caract´ erisation optimale de la structure diﬀ´ erentie lle standard de la sph` ere en terme de courbure pour (presque )toutes les dimensions. I. Les ´ enonc´ es , (French) [A sharp curvature-type characterization of the smooth spher e in all but a few dimensions. I. Statement of the theorems] C.R. Acad. Sci. Paris S´ er. I Math ., 319:6 (1994), pp. 605–607. [84] ,A sharp characterization of the smooth 4-sphere in curvature terms , Comm. Anal. Geom., 6:1 (1998), pp. 21–65. [85]C. McMullen ,Renormalization and 3-manifolds which ﬁber over the circle , Annals of Math- ematics Studies 12, Princeton U. Press, 1996. [86]W.W. Meeks, III and S.-T. Yau ,The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z., 179:2 (1982), pp. 151–168. [87]W.W. Meeks, III, L. Simon, and S.-T. Yau ,Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature , Ann. of Math., 116 (1982), pp. 621–659. [88]M. Micallef and J.D. Moore, J Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes , Ann. of Math. (2), 127 (1988), pp. 199– 227. [89]A.D. Milka ,Metric structure of some class of spaces containing straigh t lines , Ukrain. Geo- metrical. Sbornik, 4 (1967), pp. 43–48. [90]J. Milnor ,A unique factorization theorem for 3-manifolds , Amer. J. Math., 84 (1962), pp. 1– 7. [91] ,Towards the Poincar´ e conjecture and the classiﬁcation of 3-manifolds , Notices Amer. Math. Soc., 50:10 (2003), pp. 1226–1233. [92]N. Mok ,The uniformization theorem for compact K¨ ahler manifolds o f nonnegative holomor- phic bisectional curvature , J. Diﬀerential Geom., 27 (1988), pp. 179–214. [93]N. Mok, Y.T. Siu, and S.-T. Yau ,The Poincar´ e–Lelong equation on complete K¨ ahler man- ifolds, Compositio Math., 44 (1981), pp. 183–218. [94]J.W. Morgan ,On Thurston’s uniformization theorem for three-dimension al manifolds , The Smith Conjecture (Bass and Morgan, eds.), Pure Appl. Math., 112, Academic Press, (1984), pp. 37–125. [95] ,Recent progress on the Poincar´ e conjecture and the classiﬁ cation of 3-manifolds , Bull.of the A.M.S., 42:1 (2005), pp. 57–78. [96]S. Mori ,Projective manifolds with ample tangent bundles , Ann. of Math., 100 (1979), pp. 593– 606. [97]G.D. Mostow ,Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, Princeton University Press, Princeton, N.J.; Universi ty of Tokyo Press, Tokyo, 1973. [98]L. Ni ,Ancient solution to Kahler-Ricci ﬂow, arXiv:math.DG.0502494, February 2005, preprint. [99] ,A note on Perelman’s LYH inequality, arXiv:math.DG.0602337, February 2006, preprint. [100]L. Ni and L.F. Tam ,Plurisubharmonic functions and the structure of complete K ¨ ahler man- 490 H.-D. CAO AND X.-P. ZHU ifolds with nonnegative curvature , J. Diﬀerential Geom., 64:3 (2003), pp. 457–524. [101]S. Nishikawa ,Deformation of Riemannian metrics and manifolds with bound ed curvature ratios, Geometric measure theory and the calculus of variations (A rcata, Calif., 1984), pp. 343–352, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986. [102]J.-P. Otal ,Le th´ eor´ eme d’hyperbolisation pir les vari´ et´ es ﬁbr´ ee s de dimension 3 , At´ erisque 235, 1996. [103]G Perelmann ,The entropy formula for the Ricci ﬂow and its geometric appli cations , arXiv:math. DG/0211159 v1 November 11, 2002, preprint. [104] ,Ricci ﬂow with surgery on three manifolds , arXiv:math.DG/0303109 v1 March 10, 2003, preprint. [105] ,Finite extinction time to the solutions to the Ricci ﬂow on ce rtain three manifolds , arXiv:math.DG/0307245 July 17, 2003, preprint. [106]S. Peters ,Convergence of Riemannian mainfolds , Comp. Math., 62 (1987), pp. 3–16. [107]G. Prasad ,Strong rigidity of Q-rank 1lattices, Invent. Math., 21 (1973), pp. 255–286. [108]O.S. Rothaus ,Logarithmic Sobolev inequalities and the spectrum of Schr ¨odinger opreators, J. Funct. Anal., 42:1 (1981), pp. 110–120. [109]R. Schoen and S.-T. Yau ,On the structure of manifolds with positive scalar curvatur e, Manuscripta math., 28 (1979), pp. 159–183. [110] ,Existence of incompressible minimal surfaces and the topol ogy of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2), 110:1 (1979), pp. 127– 142. [111] ,Complete three dimensional manifolds with positive Ricci c urvature and scalar cur- vature , Seminor on Diﬀerential Geometry, Princeton University Pr ess, 1982, pp. 209–228. [112] ,Lectures on diﬀerential geometry , in ‘Conference Proceedings and Lecture Notes in Geometry and Topology’, 1, International Press Publicatio ns, 1994. [113]N. Sesum ,Limiting behaviour of the Ricci ﬂow, arXiv:math.DG/0402194, February 2004. [114]W.X. Shi ,Deforming the metric on complete Riemannian manifold , J. Diﬀerential Geometry, 30 (1989), pp. 223–301. [115] ,Ricci deformation of the metric on complete noncompact K¨ ah ler manifolds, Ph.D. Thesis, Harvard University, 1990. [116] ,Complete noncompact K¨ ahler manifolds with positive holom orphic bisectional cur- vature , Bull. Amer Math. Soc., 23 (1990), pp. 437–440. [117] ,Ricci ﬂow and the uniformization on complete noncompact K¨ a hler manifolds, J. Diﬀerential Geom., 45:1 (1997), pp. 94–220. [118]T. Shioya and T. Yamaguchi ,Volume collapsed three-manifolds with a lower curvature bound, Math. Ann., 333 (2005), pp. 131–155. [119]I.M. Singer ,Inﬁnitesimallly homogeneous spaces , Comm. Pure Appl. Math., 13 (1960), pp. 685–697. [120]Y.T. Siu and S.-T. Yau ,Complete K¨ ahler manifolds with non–positive curvature of faster than quadratic decay , Ann. of Math., 105 (1977), pp. 225–264. [121]M. Struwe ,Curvature ﬂows on surfaces , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1:2 (2002), pp. 247–274. [122]W.P. Thurston ,Three-dimensional manifolds, Kleinian groups and hyperbo lic geometry, Bull. Amer. Math. Soc. (N.S.), 6:3 (1982), pp. 357–381. [123] ,Hyperbolic structures on 3-manifolds. I:Deformation of acylindrical manifolds, Ann. of Math. (2), 124:2 (1986), pp. 203–246. [124] ,Hyperbolic structures on 3-manifolds II:Surface groups and 3-manifolds which ﬁber over the circle, preprint, 1988. [125] ,Three-dimensional geometry and topology, Vol. 1, Edited by Silvio Levy, Princeton Mathematical Series, 35, Princeton University Press, Prin ceton, NJ, 1997. [126]F. Waldhausen ,Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, II, Invent. Math., 3 (1967), pp. 308–333; 4 (1967), pp. 87–117. [127]X.J. Wang, and X.H. Zhu ,K¨ ahler-Ricci solitons on toric manifolds with positive ﬁr st Chern class, Adv. Math., 188:1 (2004), pp. 87–103. [128]S.-T. Yau ,Some function-theoretic properties of complete Riemannia n manifold and their applications to geometry , Indiana Univ. Math. J., 25 (1976), pp. 659–670. [129] ,The role of partial diﬀerential equations in diﬀerential ge ometry , in ‘Proceedings of the International Congress of Mathematicians’, Helsinki, 1978, pp. 237–250. [130] ,Perspectives on geometric analysis , to appear in Survey in Diﬀerential Geometry, Vol. X. (arXiv: math.DG/0602363, February, 2006) THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 491 INDEX Ticonverges to a tensor T, 267 δ-cutoﬀ surgery, 424 κ-noncollapsed, 255, 433 λ-remote, 360 L-Jacobian, 252 L-Jacobian ﬁeld, 248 L-distance function, 245 L-exponential map with parameter ¯ τ, 252 L-geodesic, 244 curve, 244 equation, 244 L-length, 244 ε-cap, 414 ε-horn, 414 ε-neck, 414 strong, 416 ε-neck of radius r, 357 ε-tube, 414 k-jet distance, 343 space, 343 (almost) maximum points, 291 a priori assumptions (with accuracy ε), 416 admissible curve, 435 ancient κ-solution, 357 solution, 233 asymptotic scalar curvature ratio, 303, 362 asymptotic volume ratio, 373 atoroidal, 482 barely admissible curve, 435 be modeled, 481 Berger’s rigidity theorem, 321 breather, 199 expanding, 199 shrinking, 199 steady, 199 canonical neighborhood assumption (with accuracy ε), 416 canonical neighborhood theorem, 396 cappedε-horn, 414 center of an evolving ε-neck, 394Cheeger’s lemma, 288 classical sphere theorems, 321 classiﬁcation of three-dimensional shrinking solitons, 384 collapsed, 338 compactness of ancient κ0-solutions, 393 conjugate heat equation, 235 converges to a marked manifold, 267 converges to an evolving marked man- ifold, 282 curvatureβ-bump, 360 degree, 308 doubleε-horn, 414 Einstein manifold, 174 metric, 174 elliptic type estimate, 391 essential component, 476 evolvingε-cap, 396 evolvingε-neck, 396 exceptional part, 338 ﬁnite bump theorem, 360 free boundary condition, 345 geometrizable, 482 gradient shrinking Ricci soliton, 384 graph manifold, 482 Gromoll-Meyer injectivity radius esti- mate, 286 Haken, 481 Hamilton’s advanced maximum princi- ple, 218 Hamilton’s compactness theorem, 285 Hamilton’s strong maximum principle, 213 Hamilton-Ivey pinching estimate, 224, 336 homogeneous manifold, 481 locally, 481 incompressible, 338 injectivity radius, 286 condition, 291 492 H.-D. CAO AND X.-P. ZHU irreducible, 476 Jacobian comparison theorem, 253 justiﬁcation of the canonical neighbor- hood assumption, 433 justiﬁcation of the pinching assump- tion, 424 K¨ ahler-Ricci ﬂow, 176 K¨ ahler-Ricci soliton expanding, 176 shrinking, 176 steady, 176 Klingenberg’s lemma, 286 Li-Yau-Hamilton estimate, 226, 230 Li-Yau-Hamilton quadratic, 230 Li-Yau-Perelman distance, 238, 250 Little Loop Lemma, 290 marked Riemannian manifold, 267 marking, 267 maximal solution, 291 Mostow type rigidity, 343 no local collapsing theorem I, 255, 256 no local collapsing theorem I′, 259 no local collapsing theorem II, 263 noncollapsing limit, 338 normalized, 398 normalized Ricci ﬂow, 307 origin, 267 Perelman’s claim, 482 Perelman’s reduced volume, 243, 252 element, 254 pinching assumption, 416 Poincar´ e conjecture, 452 prime, 481 decomposition theorem, 481 regular, 411 Ricci ﬂow, 173 Ricci ﬂow with surgery, 416 Ricci soliton expanding, 175 shrinking, 175 gradient , 175 steady , 175 Shi’s derivative estimate, 192singularity model, 292 singularity structure theorem, 399 soliton cigar, 177 steady, 175 solution ancient, 233 nonsingular, 336 standard, 420 solution becomes extinct, 417 solution develops a singularity, 267 standard capped inﬁnite cylinder, 420 support function, 220 surgery times, 416 surgically modiﬁed solution, 416 tangent cone, 219 thick-thin decomposition theorem, 473 Thurston’s geometrization conjecture, 482, 483 type I, 291–293 type II, 293 (a), 291, 294 (b), 292, 295 type III, 293 (a), 292, 296 (b), 292 universal noncollapsing, 388 Weaker Assertion, 483