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http://dx.doi.org/10.1090/gsm/077 Hamilton's Ricci Flow This page intentionally left blank Hamilton' s Ricc i Flo w Bennett Chow Peng Lu Lei Ni Graduate Studies in Mathematics Volum e 11 j^^ MtSJff l America n Mathematica l Societ y Wm^jz Scienc e Pres s Editorial Board David Cox Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) This edition is published by the American Mathematical Society under license from Science Press. 2000 Mathematics Subject Classification. Primary 53C44, 53C21, 58J35, 35K55. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-77 Library of Congress Cataloging-in-Publication Data Chow, Bennett. Hamilton's Ricci flow / Bennett Chow, Peng Lu, and Lei Ni. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 77) Includes bibliographical references and index. ISBN-13: 978-0-8218-4231-7 (alk. paper) ISBN-10: 0-8218-4231-5 (alk. paper) 1. Global differential geometry. 2. Ricci flow. 3. Riemannian manifolds. I. Lu, Peng, 1964- II. Ni, Lei, 1969- III. Title. QA670.C455 2006 516.3/62—dc22 2006043027 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 [email protected]. © 2006 by the authors. All rights reserved. Reprinted by the American Mathematical Society, 2010. 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 15 14 13 12 11 10 Contents Preface xi Acknowledgments xvii A Detailed Guide for the Reader xxi Notation and Symbols xxxv Chapter 1. Riemannian Geometry 1 §1. Introduction 1 §2. Metrics, connections, curvatures and covariant differentiation 2 §3. Basic formulas and identities in Riemannian geometry 10 §4. Exterior differential calculus and Bochner formulas 14 §5. Integration and Hodge theory 20 §6. Curvature decomposition and locally conformally flat manifolds 25 §7. Moving frames and the Gauss-Bonnet formula 32 §8. Variation of arc length, energy and area 41 §9. Geodesies and the exponential map 52 §10. Second fundamental forms of geodesic spheres 58 §11. Laplacian, volume and Hessian comparison theorems 67 §12. Proof of the comparison theorems 73 §13. Manifolds with nonnegative curvature 80 §14. Lie groups and left-invariant metrics 87 §15. Notes and commentary 89 VI Contents Chapter 2. Fundamentals of the Ricci Flow Equation 95 §1. Geometric flows and geometrization 96 §2. Ricci flow and the evolution of scalar curvature 98 §3. The maximum principle for heat-type equations 100 §4. The Einstein-Hilbert functional 104 §5. Evolution of geometric quantities 108 §6. DeTurck's trick and short time existence 113 §7. Reaction-diffusion equation for the curvature tensor 119 §8. Notes and commentary 123 Chapter 3. Closed 3-manifolds with Positive Ricci Curvature 127 §1. Hamilton's 3-manifolds with positive Ricci curvature theorem 127 §2. The maximum principle for tensors 128 §3. Curvature pinching estimates 131 §4. Gradient bounds for the scalar curvature 136 §5. Curvature tends to constant 140 §6. Exponential convergence of the normalized flow 142 §7. Notes and commentary 149 Chapter 4. Ricci Solitons and Special Solutions 153 §1. Gradient Ricci solitons 154 §2. Gaussian and cylinder solitons 157 §3. Cigar steady soliton 159 §4. Rosenau solution 162 §5. An expanding soliton 164 §6. Bryant soliton 167 §7. Homogeneous solutions 169 §8. The isometry group 175 §9. Notes and commentary 176 Chapter 5. Isoperimetric Estimates and No Local Collapsing 181 §1. Sobolev and logarithmic Sobolev inequalities 181 §2. Evolution of the length of a geodesic 186 §3. Isoperimetric estimate for surfaces 188 §4. Perelman's no local collapsing theorem 190 §5. Geometric applications of no local collapsing 198 §6. 3-manifolds with positive Ricci curvature revisited 206 Contents vn §7. Isoperimetric estimate for 3-dimensional Type I solutions 208 §8. Notes and commentary 211 Chapter 6. Preparation for Singularity Analysis 213 §1. Derivative estimates and long time existence 213 §2. Proof of Shi's local first and second derivative estimates 218 §3. Cheeger-Gromov-type compactness theorem for Ricci flow 233 §4. Long time existence of solutions with bounded Ricci curvature 237 §5. The Hamilton-Ivey curvature estimate 240 §6. Strong maximum principles and metric splitting 245 §7. Rigidity of 3-manifolds with nonnegative curvature 248 §8. Notes and commentary 250 Chapter 7. High-dimensional and Noncompact Ricci Flow 253 §1. Spherical space form theorem of Huisken-Margerin-Nishikawa 254 §2. 4-manifolds with positive curvature operator 259 §3. Manifolds with nonnegative curvature operator 263 §4. The maximum principle on noncompact manifolds 272 §5. Complete solutions of the Ricci flow on noncompact manifolds 279 §6. Notes and commentary 286 Chapter 8. Singularity Analysis 291 §1. Singularity dilations and types 292 §2. Point picking and types of singularity models 297 §3. Geometric invariants of ancient solutions 307 §4. Dimension reduction 316 §5. Notes and commentary 326 Chapter 9. Ancient Solutions 327 §1. Classification of ancient solutions on surfaces 328 §2. Properties of ancient solutions that relate to their type 338 §3. Geometry at infinity of gradient Ricci solitons 353 §4. Injectivity radius of steady gradient Ricci solitons 364 §5. Towards a classification of 3-dimensional ancient solutions 368 §6. Classification of 3-dimensional shrinking Ricci solitons 375 §7. Summary and open problems 388 Chapter 10. Differential Harnack Estimates 391 viii Contents §1. Harnack estimates for the heat and Laplace equations 392 §2. Harnack estimate on surfaces with \ > 0 397 §3. Linear trace and interpolated Harnack estimates on surfaces 401 §4. Hamilton's matrix Harnack estimate for the Ricci flow 405 §5. Proof of the matrix Harnack estimate 410 §6. Harnack and pinching estimates for linearized Ricci flow 415 §7. Notes and commentary 420 Chapter 11. Space-time Geometry 425 §1. Space-time solution to the Ricci flow for degenerate metrics 426 §2. Space-time curvature is the matrix Harnack quadratic 433 §3. Potentially infinite metrics and potentially infinite dimensions 434 §4. Renormalizing the space-time length yields the ^-length 452 §5. Space-time DeTurck's trick and fixing the measure 453 §6. Notes and commentary 456 Appendix A. Geometric Analysis Related to Ricci Flow 461 §1. Compendium of inequalities 461 §2. Comparison theory for the heat kernel 463 §3. Green's function 465 §4. The Liouville theorem revisited 466 §5. Eigenvalues and eigenfunctions of the Laplacian 467 §6. The determinant of the Laplacian 476 §7. Parametrix for the heat equation 485 §8. Monotonicity for harmonic functions and maps 492 §9. Bieberbach theorem 494 §10. Notes and commentary 500 Appendix B. Analytic Techniques for Geometric Flows 503 §1. Riemannian surfaces 503 §2. Kazdan-Warner-type identities and solitons 516 §3. Andrews' Poincare-type inequality 517 §4. The Yamabe flow and Aleksandrov reflection 520 §5. The cross curvature flow 528 §6. Time derivative of the sup function 531 §7. Notes and commentary 532 Appendix S. Solutions to Selected Exercises 535 Contents IX Bibliography 573 Index 603 This page intentionally left blank Preface About the book. Ricci flow is a geometric and analytic evolution equa• tion which we believe is related to physical reality. One of the underlying principles of science is unity. In Ricci flow we see the unity of geometry and analysis. It is also expected to exhibit unity with low-dimensional topology. In this book we emphasize the more geometric and analytic aspects of Ricci flow rather than the topological aspects. We also attempt to convey some of the relations and formal similarities between Ricci flow and other geometric flows such as mean curvature flow. The interaction of techniques and ideas between Ricci flow and other geometric flows is a two-way street. So we hope the reader with a more general interest in geometric flows will benefit from the usefulness of applying ideas originating in Ricci flow to the study of other geometric flows. We have not aimed at completeness, even in the realm of the limited material that we cover. A more extensive coverage of the subject of Ricci flow is planned in the book by Dan Knopf and one of the authors [163] and its multi-authored sequel [153]. At places we follow the informal style of lecture notes and have at• tempted to cover some of the basic material in a relatively direct and effi• cient way. At the same time we take the opportunity to expose the reader to techniques, some of which lie outside of the subject of Ricci flow per se, which he or she may find useful in pursuing research in Ricci flow. So, metaphorically speaking, this book is a hybrid between rushing to work in the morning on a cold and blistery winter day and a casual stroll through the park on a warm and sunny midsummer afternoon. As much as possible, we have attempted to construct the book so that the chapters, and in some cases, individual sections, are relatively independent. In this way we hope XI Xll Preface that the book may be used as both an introduction and as a reference.
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