**Mathematical Surveys and Monographs**

Volume: 110;
2004;
325 pp;
Hardcover

MSC: Primary 53; 58; 35; 57;

Print ISBN: 978-0-8218-3515-9

Product Code: SURV/110

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-1-4704-1337-8
Product Code: SURV/110.E**

List Price: $97.00

Individual Member Price: $77.60

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#### Supplemental Materials

# The Ricci Flow: An Introduction

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*Bennett Chow; Dan Knopf*

The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to “flow” a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics.

Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program.

The Ricci flow method is now central to our understanding of the geometry and topology of manifolds. This book is an introduction to that program and to its connection to Thurston's geometrization conjecture.

The authors also provide a “Guide for the hurried reader”, to help readers wishing to develop, as efficiently as possible, a nontechnical appreciation of the Ricci flow program for 3-manifolds, i.e., the so-called “fast track”.

The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds.

*The Ricci Flow*

#### Table of Contents

# Table of Contents

## The Ricci Flow: An Introduction

- Contents iii4 free
- Preface vii8 free
- Chapter 1. The Ricci flow of special geometries 114 free
- 1. Geometrization of three-manifolds 215
- 2. Model geometries 417
- 3. Classifying three-dimensional maximal model geometries 619
- 4. Analyzing the Ricci flow of homogeneous geometries 821
- 5. The Ricci flow of a geometry with maximal isotropy SO (3) 1124
- 6. The Ricci flow of a geometry with isotropy SO (2) 1528
- 7. The Ricci flow of a geometry with trivial isotropy 1730
- Notes and commentary 1932

- Chapter 2. Special and limit solutions 2134
- Chapter 3. Short time existence 6780
- Chapter 4. Maximum principles 93106
- Chapter 5. The Ricci flow on surfaces 105118
- 1. The effect of a conformal change of metric 106119
- 2. Evolution of the curvature 109122
- 3. How Ricci solitons help us estimate the curvature from above 111124
- 4. Uniqueness of Ricci solitons 116129
- 5. Convergence when χ(M[sup(2)]) < 0 120133
- 6. Convergence when χ(M[sup(2)]) = 0 123136
- 7. Strategy for the case that χ(M[sup(2)] > 0) 128141
- 8. Surface entropy 133146
- 9. Uniform upper bounds for R and |∇R| 137150
- 10. Differential Harnack estimates of LYH type 143156
- 11. Convergence when R(·,0) > 0 148161
- 12. A lower bound for the injectivity radius 149162
- 13. The case that R(·,0) changes sign 153166
- 14. Monotonicity of the isoperimetric constant 156169
- 15. An alternative strategy for the case χ(M[sup(2)] > 0) 165178
- Notes and commentary 171184

- Chapter 6. Three-manifolds of positive Ricci curvature 173186
- 1. The evolution of curvature under the Ricci flow 174187
- 2. Uhlenbeck's trick 180193
- 3. The structure of the curvature evolution equation 183196
- 4. Reduction to the associated ODE system 187200
- 5. Local pinching estimates 189202
- 6. The gradient estimate for the scalar curvature 194207
- 7. Higher derivative estimates and long-time existence 200213
- 8. Finite-time blowup 209222
- 9. Properties of the normalized Ricci flow 212225
- 10. Exponential convergence 218231
- Notes and commentary 221234

- Chapter 7. Derivative estimates 223236
- Chapter 8. Singularities and the limits of their dilations 233246
- Chapter 9. Type I singularities 253266
- Appendix A. The Ricci calculus 279292
- 1. Component representations of tensor fields 279292
- 2. First-order differential operators on tensors 280293
- 3. First-order differential operators on forms 283296
- 4. Second-order differential operators 284297
- 5. Notation for higher derivatives 285298
- 6. Commuting covariant derivatives 286299
- Notes and commentary 286299

- Appendix B. Some results in comparison geometry 287300
- Bibliography 317330
- Index 323336

#### Readership

Graduate students and research mathematicians interested in geometric analysis.

#### Reviews

Well written ... topics are well-motivated and presented with clarity and insight ... proofs are clear yet detailed so that the material is accessible to a wide audience as well as specialists. Gaps in arguments in the literature are filled in.

-- Zentralblatt MATH