The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic Aspects
About this Title
Bennett Chow, University of California, San Diego, La Jolla, CA, Sun-Chin Chu, National Chung Cheng University, Chia-Yi, Taiwan, David Glickenstein, University of Arizona, Tucson, AZ, Christine Guenther, Pacific University, Forest Grove, OR, James Isenberg, University of Oregon, Eugene, OR, Tom Ivey, College of Charleston, Charleston, SC, Dan Knopf, University of Texas, Austin, Austin, TX, Peng Lu, University of Oregon, Eugene, OR, Feng Luo, Rutgers University, Piscataway, NJ and Lei Ni, University of California, San Diego, La Jolla, CA
Publication: Mathematical Surveys and Monographs
Publication Year: 2010; Volume 163
ISBNs: 978-0-8218-4661-2 (print); 978-1-4704-1390-3 (online)
MathSciNet review: MR2604955
MSC: Primary 53C44; Secondary 35K08, 35K55
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.
Graduate students and research mathematicians interested in geometric analysis, Ricci flow, Perelman's work on Poincaré.
Table of Contents
- 1. Entropy, $\mu $-invariant, and finite time singularities
- 2. Geometric tools and point picking methods
- 3. Geometric properties of $\kappa $-solutions
- 4. Compactness of the space of $\kappa $-solutions
- 5. Perelman’s pseudolocality theorem
- 6. Tools used in proof of pseudolocality
- 7. Heat kernel for static metrics
- 8. Heat kernel for evolving metrics
- 9. Estimates of the heat equation for evolving metrics
- 10. Bounds for the heat kernel for evolving metrics
- 11. Elementary aspects of metric geometry
- 12. Convex functions on Riemannian manifolds
- 13. Asymptotic cones and Sharafutdinov retraction
- 14. Solutions to selected exercises