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The Ricci Flow: Techniques and Applications http://dx.doi.org/10.1090/surv/135 The Ricci Flow: Techniques and Applications Part I: Geometric Aspects Mathematical Surveys and Monographs Volume 135 The Ricci Flow: Techniques and Applications Part I: Geometric Aspects Bennett Chow Sun-Chin Chu David Glickenstein Christine Guenther James Isenberg Tom Ivey Dan Knopf Peng Lu Feng Luo Lei Ni American Mathematical Society *WDED EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 53C44, 53C25, 58J35, 35K55, 35K05. For additional information and updates on this book, visit www.ams.org/bookpages/surv-135 ISBN 978-0-8218-3946-1 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams.org. © 2007 by Bennett Chow. All rights reserved. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 Contents Preface ix What this book is about ix Highlights of Part I xi Acknowledgments xiii Contents of Part I of Volume Two xvii Chapter 1. Ricci Solitons 1 1. General solitons and their canonical forms 2 2. Differentiating the soliton equation — local and global analysis 6 3. Warped products and 2-dimensional solitons 11 4. Constructing the Bryant steady soliton 17 5. Rotationally symmetric expanding solitons 26 6. Homogeneous expanding solitons 32 7. When breathers and solitons are Einstein 41 8. Perelman's energy and entropy in relation to Ricci solitons 44 9. Buscher duality transformation of warped product solitons 46 10. Summary of results and open problems on Ricci solitons 50 11. Notes and commentary 52 Chapter 2. Kahler-Ricci Flow and Kahler-Ricci Solitons 55 1. Introduction to Kahler manifolds 55 2. Connection, curvature, and covariant differentiation 62 3. Existence of Kahler-Einstein metrics 70 4. Introduction to the Kahler-Ricci flow 74 5. Existence and convergence of the Kahler-Ricci flow 80 6. Survey of some results for the Kahler-Ricci flow 95 7. Examples of Kahler-Ricci solitons 97 8. Kahler-Ricci flow with nonnegative bisectional curvature 103 9. Matrix differential Harnack estimate for the Kahler-Ricci flow 109 10. Linear and interpolated differential Harnack estimates 118 11. Notes and commentary 124 Chapter 3. The Compactness Theorem for Ricci Flow 127 1. Introduction and statements of the compactness theorems 127 2. Convergence at all times from convergence at one time 132 3. Extensions of Hamilton's compactness theorem 138 vi CONTENTS 4. Applications of Hamilton's compactness theorem 142 5. Notes and commentary 148 Chapter 4. Proof of the Compactness Theorem 149 1. Outline of the proof 149 2. Approximate isometries, compactness of maps, and direct limits 150 3. Construction of good coverings by balls 158 4. The limit manifold (M^goo) 165 5. Center of mass and nonlinear averages 175 6. Notes and commentary 187 Chapter 5. Energy, Monotonicity, and Breathers 189 1. Energy, its first variation, and the gradient flow 190 2. Monotonicity of energy for the Ricci flow 197 3. Steady and expanding breather solutions revisited 203 4. Classical entropy and Perelman's energy 214 5. Notes and commentary 219 Chapter 6. Entropy and No Local Collapsing 221 1. The entropy functional W and its monotonicity 221 2. The functionals fi and v 235 3. Shrinking breathers are shrinking gradient Ricci solitons 242 4. Logarithmic Sobolev inequality 246 5. No finite time local collapsing: A proof of Hamilton's little loop conjecture 251 6. Improved version of no local collapsing and diameter control 264 7. Some further calculations related to T and W 273 8. Notes and commentary 284 Chapter 7. The Reduced Distance 285 1. The ^-length and distance for a static metric 286 2. The ^-length and the L-distance 288 3. The first variation of £-length and existence of ^-geodesies 296 4. The gradient and time-derivative of the L-distance function 306 5. The second variation formula for C and the Hessian of L 312 6. Equations and inequalities satisfied by L and £ 322 7. The ^-function on Einstein solutions and Ricci solitons 335 8. £-Jacobi fields and the /^-exponential map 345 9. Weak solution formulation 363 10. Notes and commentary 379 Chapter 8. Applications of the Reduced Distance 381 1. Reduced volume of a static metric 381 2. Reduced volume for Ricci flow 386 3. A weakened no local collapsing theorem via the monotonicity of the reduced volume 399 4. Backward limit of ancient ^-solution is a shrinker 406 CONTENTS vii 5. Perelman's Riemannian formalism in potentially infinite dimensions 417 6. Notes and commentary 432 Chapter 9. Basic Topology of 3-Manifolds 433 1. Essential 2-spheres and irreducible 3-manifolds 433 2. Incompressible surfaces and the geometrization conjecture 435 3. Decomposition theorems and the Ricci flow 439 4. Notes and commentary 442 Appendix A. Basic Ricci Flow Theory 445 1. Riemannian geometry 445 2. Basic Ricci flow 456 3. Basic singularity theory for Ricci flow 465 4. More Ricci flow theory and ancient solutions 470 5. Classical singularity theory 474 Appendix B. Other Aspects of Ricci Flow and Related Flows 477 1. Convergence to Ricci solitons 477 2. The mean curvature flow 482 3. The cross curvature flow 490 4. Notes and commentary 500 Appendix C. Glossary 501 Bibliography 513 Index 531 Preface Trinity: It's the question that drives us, Neo. It's the question that brought you here. You know the question just as I did. Neo: What is the Matrix? ... Morpheus: Do you want to know what it is? The Matrix is everywhere.... Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. - From the movie "The Matrix". What this book is about This is the sequel to the book "The Ricci Flow: An Introduction" by two of the authors [108]. In the previous volume (henceforth referred to as Volume One) we laid some of the foundations for the study of Richard Hamilton's Ricci flow. The Ricci flow is an evolution equation which deforms Riemannian metrics by evolving them in the direction of minus the Ricci tensor. It is like a heat equation and tries to smooth out initial metrics. In some cases one can exhibit global existence and convergence of the Ricci flow. A striking example of this is the main result presented in Chapter 6 of Volume One: Hamilton's topological classification of closed 3-manifolds with positive Ricci curvature as spherical space forms. The idea of the proof is to show, for any initial metric with positive Ricci curvature, the normalized Ricci flow exists for all time and converges to a constant curvature metric as time approaches infinity. Note that on any closed 2-dimensional manifold, the normalized Ricci flow exists for all time and converges to a constant curvature metric. Many of the techniques used in Hamilton's original work in dimension 2 have influenced the study of the Ricci flow in higher dimensions. In this respect, of special note is Hamilton's 'meta-principle' of considering geometric quantities which either vanish or are constant on gradient Ricci solitons. It is perhaps generally believed that the Ricci flow tries to make met­ rics more homogeneous and isotropic. However, for general initial metrics on closed manifolds, singularities may develop under the Ricci flow in di­ mensions as low as 3.1 In Volume One we began to set up the study of singularities by discussing curvature and derivative of curvature estimates, looking at how generally dilations are done in all dimensions, and studying For noncompact manifolds, finite time singularities may even occur in dimension 2. ix x PREFACE aspects of singularity formation in dimension 3. In this volume, we continue the study of the fundamental properties of the Ricci flow with particular emphasis on their application to the study of singularities. We pay particu­ lar attention to dimension 3, where we describe some aspects of Hamilton's and Perelman's nearly complete classification of the possible singularities.2 As we saw in Volume One, Ricci solitons (i.e., self-similar solutions), dif­ ferential Harnack inequalities, derivative estimates, compactness theorems, maximum principles, and injectivity radius estimates play an important role in the study of the Ricci flow. The maximum principle was used extensively in the 3-dimensional results we presented. Some of the other techniques were presented only in the context of the Ricci flow on surfaces. In this volume we take a more detailed look at these general topics and also describe some of the fundamental new tools of Perelman which almost complete Hamilton's partial classification of singularities in dimension 3. In particular, we discuss Perelman's energy, entropy, reduced distance, and some applications. Much of Perelman's work is independent of dimension and leads to a new under­ standing of singularities. It is difficult to overemphasize the importance of the reduced distance function, which is a space-time distance-like function (not necessarily nonnegative!) which is intimately tied to the geometry of solutions of the Ricci flow and the understanding of forming singularities.
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