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Notes on Perelman's Papers Geometry & Topology 12 (2008) 2587–2858 msp Notes on Perelman’s papers BRUCE KLEINER JOHN LOTT These are detailed notes on Perelman’s papers “The entropy formula for the Ricci flow and its geometric applications”[46] and “Ricci flow with surgery on three-manifolds” [47]. 57M40, 57M50; 53C21, 53C44 1 Introduction These are notes on Perelman’s papers “The Entropy Formula for the Ricci Flow and its Geometric Applications”[46] and “Ricci Flow with Surgery on Three-Manifolds’[47]. In these two remarkable preprints, which were posted on the arXiv in 2002 and 2003, Grisha Perelman announced a proof of the Poincare´ Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow approach of Hamilton. Perelman’s proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in[46] and[47], which contain Perelman’s arguments for the Geometrization Conjecture. Among other things, we cover the construction of the Ricci flow with surgery of[47]. We also discuss the long-time behavior of the Ricci flow with surgery, which is needed for the full Geometrization Conjecture. The papers of Colding and Minicozzi[23; 24] and Perelman[48], which are not covered in these notes, each provide a shortcut in the case of the Poincare´ Conjecture. Namely, these papers show that if the initial manifold is simply-connected then the Ricci flow with surgery becomes extinct in a finite time, thereby removing the issue of the long-time behavior. Combining this claim with the proof of existence of Ricci flow with surgery gives the shortened proof in the simply-connected case. These notes are intended for readers with a solid background in geometric analysis. Good sources for background material on Ricci flow are Chow and Knopf[21], Chow, Lu and Ni[22], Hamilton[30] and Topping[60]. The notes are self-contained but are designed to be read along with[46; 47]. For the most part we follow the format of[46; 47] and use the section numbers of[46; 47] to label our sections. We have done this in Published: 6 November 2008 DOI: 10.2140/gt.2008.12.2587 2588 Bruce Kleiner and John Lott order to respect the structure of[46; 47] and to facilitate the use of the present notes as a companion to[46; 47]. In some places we have rearranged Perelman’s arguments or provided alternative arguments, but we have refrained from an overall reorganization. Besides providing details for Perelman’s proofs, we have included some expository material in the form of overviews and appendices. Section 3 contains an overview of the Ricci flow approach to geometrization of 3–manifolds. Sections 4 and 57 contain overviews of[46] and[47], respectively. The appendices discuss some background material and techniques that are used throughout the notes. Regarding the proofs, the papers[46; 47] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader. (Some of the mistakes in[46] were corrected in[47].) We did not find any serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman. We will refer to Section X.Y of[46] as “I.X.Y”, and Section X.Y of[47] as “II.X.Y”. A reader may wish to start with the overviews, which explain the logical structure of the arguments and the interrelations between the sections. It may also be helpful to browse through the appendices before delving into the main body of the material. These notes have gone through various versions, which were posted on our web site [36]. An initial version with notes on[46] was posted in June 2003. A version covering [46; 47] was posted in September 2004. After the May 2006 version of these notes was posted on the arXiv, expositions of Perelman’s work appeared in Cao and Zhu[15] and Morgan and Tian[42]. This is the 20 February 2013 version of the notes. Acknowledgements In the preparation of the September 2004 version of our notes, we benefited from a workshop on Perelman’s surgery procedure that was held in August 2004, at Princeton. We thank the participants of that meeting, as well as the Clay Mathematics Institute for supporting it. We are grateful for comments and corrections that we have received regarding earlier versions of these notes. We especially thank Gerard´ Besson for comments on Theorem 13.3 and Lemma 80.4, Peng Lu for comments on Lemma 74.1, Bernhard Leeb for discussions on the issue of uniqueness for the standard solution and Richard Bamler for comments on Proposition 84.1 and Proposition 84.2. Along with these people we thank Mike Anderson, Albert Chau, Ben Chow, Xianzhe Dai, Jonathan Dinkelbach, Sylvain Maillot, John Morgan, Joan Porti, Gang Tian, Peter Topping, Bing Wang, Guofang Wei, Hartmut Weiss, Burkhard Wilking, Jon Wolfson, Rugang Ye and Maciej Zworski. We thank the referees for their detailed and helpful comments on this paper. This research was supported by NSF grants DMS-0306242 and DMS-0204506, and the Clay Mathematics Institute. Geometry & Topology, Volume 12 (2008) Notes on Perelman’s papers 2589 Contents 1 Introduction 2587 2 A reading guide 2593 3 An overview of the Ricci flow approach to 3–manifold geometrization 2593 3.1 The definition of Ricci flow, and some basic properties 2594 3.3 A rough outline of the Ricci flow proof of the Poincare´ Conjecture 2595 3.8 Outline of the proof of the Geometrization Conjecture 2597 3.13 3.4 and the structure of singularities 2598 3.18 The proof of Claims 3.5 and 3.6 2601 4 Overview of ‘The Entropy Formula for the Ricci Flow and its Geometric Applications’ 2602 4.1 Summary of I.1–I.6 2603 4.2 Summary of I.7–I.10 2604 4.4 Summary of I.11 2605 4.5 Summary of I.12–I.13 2606 5 I.1.1: The F –functional and its monotonicity 2608 6 Basic example for I.1 2612 7 I.2.2: The –invariant and its applications 2613 8 I.2.3: The rescaled –invariant 2615 9 I.2.4: Gradient steady solitons on compact manifolds 2617 10 Ricci flow as a gradient flow 2618 11 The W –functional 2619 12 I.3.1: Monotonicity of the W –functional 2620 13 I.4: The no local collapsing theorem I 2623 14 I.5: The W –functional as a time derivative 2626 15 I.7: Overview of reduced length and reduced volume 2627 16 Basic example for I.7 2628 17 Remarks about L–Geodesics and L exp 2629 18 I.(7.3)–(7.6): First derivatives of L 2632 19 I.(7.7): Second variation of L 2634 20 I.(7.8)–(7.9): Hessian bound for L 2637 21 I.(7.10): The Laplacian of L 2638 22 I.(7.11): Estimates on L–Jacobi fields 2639 Geometry & Topology, Volume 12 (2008) 2590 Bruce Kleiner and John Lott 23 Monotonicity of the reduced volume V 2639 24 I.(7.15): A differential inequality for L 2641 25 I.7.2: Estimates on the reduced length 2643 26 I.7.3: The no local collapsing theorem II 2644 27 I.8.3: Length distortion estimates 2648 28 I.8.2: No local collapsing propagates forward in time and to larger scales 2651 29 I.9: Perelman’s differential Harnack inequality 2653 30 The statement of the pseudolocality theorem 2658 31 Claim 1 of I.10.1: A point selection argument 2659 32 Claim 2 of I.10.1: Getting parabolic regions 2660 33 Claim 3 of I.10.1: An upper bound on the integral of v 2661 34 Theorem I.10.1: Proof of the pseudolocality theorem 2663 35 I.10.2: The volumes of future balls 2667 36 I.10.4: –noncollapsing at future times 2668 37 I.10.5: Diffeomorphism finiteness 2668 38 I.11.1: –solutions 2669 39 I.11.2: Asymptotic solitons 2670 40 I.11.3: Two dimensional –solutions 2673 41 I.11.4: Asymptotic scalar curvature and asymptotic volume ratio 2674 42 In a –solution, the curvature and the normalized volume control each other 2678 43 An alternate proof of 40.1 using 41.13 and 42.1 2680 44 I.11.5: A volume bound 2682 45 I.11.6: Curvature bounds for Ricci flow solutions with nonnegative curvature operator, assuming a lower volume bound 2683 46 I.11.7: Compactness of the space of three-dimensional –solutions 2686 47 I.11.8: Necklike behavior at infinity of a three-dimensional –solution – weak version 2688 48 I.11.8: Necklike behavior at infinity of a three-dimensional –solution – strong version 2690 49 More properties of –solutions 2692 50 I.11.9: Getting a uniform value of 2693 51 II.1.2: Three-dimensional noncompact –noncollapsed gradient shrinkers are standard 2694 52 I.12.1: Canonical neighborhood theorem 2698 Geometry & Topology, Volume 12 (2008) Notes on Perelman’s papers 2591 53 I.12.2: Later scalar curvature bounds on bigger balls from curvature and volume bounds 2707 54 I.12.3: Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds 2710 55 I.12.4: Small balls with strongly negative curvature are volume-collapsed 2713 56 I.13.1: Thick-thin decomposition for nonsingular flows 2714 57 Overview of ‘Ricci Flow with Surgery on Three-Manifolds’ 2716 57.1 Summary of I.1–III.3 2716 57.2 Summary of II.4–II.5 2717 57.3 Summary of II.6–II.8 2720 58 II: Notation and terminology 2721 59 II.1: Three-dimensional –solutions 2724 60 II.2: Standard solutions 2728 61 Claim 2 of II.2: The blow-up time for a standard solution is 1 2730 Ä 62 Claim 4 of II.2: The blow-up time of a standard solution is 1 2731 63 Claim 5 of II.2: Canonical neighborhood property for standard solutions 2732 64 Compactness of the space of standard solutions 2733 65 Claim 1 of II.2: Rotational symmetry of standard solutions 2734 66 Claim 3 of II.2: Uniqueness of the standard solution 2736 67 II.3: Structure at the first singularity time 2739 68 Ricci flow with surgery: the general setting 2745 69 II.4.1: A priori assumptions 2749 70 II.4.2: Curvature bounds from the a priori assumptions 2751 71 II.4.3:
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