**Mathematical Surveys and Monographs**

Volume: 135;
2007;
536 pp;
Hardcover

MSC: Primary 53; 58; 35;

Print ISBN: 978-0-8218-3946-1

Product Code: SURV/135

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-1362-0
Product Code: SURV/135.E**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

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#### Supplemental Materials

# The Ricci Flow: Techniques and Applications: Part I: Geometric Aspects

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*Bennett Chow; Sun-Chin Chu; David Glickenstein; Christine Guenther; James Isenberg; Tom Ivey; Dan Knopf; Peng Lu; Feng Luo; Lei Ni*

This book gives a presentation of topics in Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject. The authors have aimed at presenting technical material in a clear and detailed manner. In this volume, geometric aspects of the theory have been emphasized. The book presents the theory of Ricci solitons, Kähler–Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local collapsing, Perelman's reduced distance function and applications to ancient solutions, and a primer of 3-manifold topology. Various technical aspects of Ricci flow have been explained in a clear and detailed manner. The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to Perelman's work and explains technical aspects of Ricci flow useful for singularity analysis. Throughout, there are appropriate references so that the reader may further pursue the statements and proofs of the various results.

#### Readership

Graduate students and research mathematicians interested in geometric analysis, specifically, the use of Ricci flow to study the geometry and topology of three-dimensional manifolds and Perelman's methods for proving the Poincaré conjecture.

#### Reviews & Endorsements

...this book is presented in a readable style, with notes and commentary concluding each chapter aiding its readability...

-- James McCoy for Mathematical Reviews

The book is meant to be a text book as well as a reference book and it includes exercises as well as the solutions for some of them. In the first part of this volume the authors took great care in making this a self contained book which compiling a great amount of facts related to the Ricci flow, organizing the information in a very clear way giving very often information on the content of each chapter and relating it to other parts of the book.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## The Ricci Flow: Techniques and Applications: Part I: Geometric Aspects

- Contents v6 free
- Preface ix10 free
- Contents of Part I of Volume Two xvii18 free
- Chapter 1. Ricci Solitons 126 free
- 1. General solitons and their canonical forms 227
- 2. Differentiating the soliton equation — local and global analysis 631
- 3. Warped products and 2-dimensional solitons 1136
- 4. Constructing the Bryant steady soliton 1742
- 5. Rotationally symmetric expanding solitons 2651
- 6. Homogeneous expanding solitons 3257
- 7. When breathers and solitons are Einstein 4166
- 8. Perelman's energy and entropy in relation to Ricci solitons 4469
- 9. Buscher duality transformation of warped product solitons 4671
- 10. Summary of results and open problems on Ricci solitons 5075
- 11. Notes and commentary 5277

- Chapter 2. Kähler–Ricci Flow and Kähler–Ricci Solitons 5580
- 1. Introduction to Kähler manifolds 5580
- 2. Connection, curvature, and covariant differentiation 6287
- 3. Existence of Kähler–Einstein metrics 7095
- 4. Introduction to the Kähler–Ricci flow 7499
- 5. Existence and convergence of the Kähler–Ricci flow 80105
- 6. Survey of some results for the Kähler–Ricci flow 95120
- 7. Examples of Kähler–Ricci solitons 97122
- 8. Kähler–Ricci flow with nonnegative bisectional curvature 103128
- 9. Matrix differential Harnack estimate for the Kähler–Ricci flow 109134
- 10. Linear and interpolated differential Harnack estimates 118143
- 11. Notes and commentary 124149

- Chapter 3. The Compactness Theorem for Ricci Flow 127152
- Chapter 4. Proof of the Compactness Theorem 149174
- Chapter 5. Energy, Monotonicity, and Breathers 189214
- Chapter 6. Entropy and No Local Collapsing 221246
- 1. The entropy functional W and its monotonicity 221246
- 2. The functionals μand v 235260
- 3. Shrinking breathers are shrinking gradient Ricci solitons 242267
- 4. Logarithmic Sobolev inequality 246271
- 5. No finite time local collapsing: A proof of Hamilton's little loop conjecture 251276
- 6. Improved version of no local collapsing and diameter control 264289
- 7. Some further calculations related to F and W 273298
- 8. Notes and commentary 284309

- Chapter 7. The Reduced Distance 285310
- 1. The L-length and distance for a static metric 286311
- 2. The L-length and the L-distance 288313
- 3. The first variation of L-length and existence of L-geodesies 296321
- 4. The gradient and time- derivative of the L-distance function 306331
- 5. The second variation formula for L and the Hessian of L 312337
- 6. Equations and inequalities satisfied by L and l 322347
- 7. The l-function on Einstein solutions and Ricci solitons 335360
- 8. L-Jacobi fields and the L-exponential map 345370
- 9. Weak solution formulation 363388
- 10. Notes and commentary 379404

- Chapter 8. Applications of the Reduced Distance 381406
- 1. Reduced volume of a static metric 381406
- 2. Reduced volume for Ricci flow 386411
- 3. A weakened no local collapsing theorem via the monotonicity of the reduced volume 399424
- 4. Backward limit of ancient k-solution is a shrinker 406431
- 5. Perelman's Riemannian formalism in potentially infinite dimensions 417442
- 6. Notes and commentary 432457

- Chapter 9. Basic Topology of 3-Manifolds 433458
- Appendix A. Basic Ricci Flow Theory 445470
- Appendix B. Other Aspects of Ricci Flow and Related Flows 477502
- Appendix C. Glossary 501526
- Bibliography 513538
- Index 531556