**Mathematical Surveys and Monographs**

Volume: 163;
2010;
517 pp;
Hardcover

MSC: Primary 53; 58; 35;

Print ISBN: 978-0-8218-4661-2

Product Code: SURV/163

List Price: $113.00

Individual Member Price: $90.40

**Electronic ISBN: 978-1-4704-1390-3
Product Code: SURV/163.E**

List Price: $113.00

Individual Member Price: $90.40

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#### Supplemental Materials

# The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic 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*

The Ricci flow uses methods from analysis to study the geometry and topology of manifolds. With the third part of their volume on techniques and applications of the theory, the authors give a presentation of Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject, with an emphasis on the geometric and analytic aspects.

The topics include Perelman's entropy functional, point picking methods, aspects of Perelman's theory of \(\kappa\)-solutions including the \(\kappa\)-gap theorem, compactness theorem and derivative estimates, Perelman's pseudolocality theorem, and aspects of the heat equation with respect to static and evolving metrics related to Ricci flow. In the appendices, we review metric and Riemannian geometry including the space of points at infinity and Sharafutdinov retraction for complete noncompact manifolds with nonnegative sectional curvature. As in the previous volumes, the authors have endeavored, as much as possible, to make the chapters independent of each other.

The book makes advanced material accessible to graduate students and nonexperts. It includes a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. The authors give the appropriate references so that the reader may further pursue the statements and proofs of the various results.

#### Table of Contents

# Table of Contents

## The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic Aspects

- Contents v6 free
- Preface ix10 free
- Contents of Part III of Volume Two xiii14 free
- Notation and Symbols xvii18 free
- Chapter 17. Entropy, $\mu$-invariant, and Finite Time Singularities 122 free
- Chapter 18. Geometric Tools and Point Picking Methods 3960
- Chapter 19. Geometric Properties of $\kappa$-Solutions 79100
- Chapter 20. Compactness of the Space of $\kappa$-Solutions 123144
- Chapter 21. Perelman's Pseudolocality Theorem 157178
- Chapter 22. Tools Used in Proof of Pseudolocality 183204
- Chapter 23. Heat Kernel for Static Metrics 215236
- Chapter 24. Heat Kernel for Evolving Metrics 265286
- Chapter 25. Estimates of the Heat Equation for Evolving Metrics 305326
- Chapter 26. Bounds for the Heat Kernel for Evolving Metrics 333354
- Appendix G. Elementary Aspects of Metric Geometry 387408
- Appendix H. Convex Functions on Riemannian Manifolds 413434
- Appendix I. Asymptotic Cones and Sharafutdinov Retraction 457478
- Appendix J. Solutions to Selected Exercises 497518
- Bibliography 503524
- Index 513534 free

#### Readership

Graduate students and research mathematicians interested in geometric analysis, Ricci flow, Perelman's work on Poincaré.