The Ricci Flow: An Introduction
About this Title
Bennett Chow, University of California, San Diego, CA and Dan Knopf, University of Texas, Austin, TX
Publication: Mathematical Surveys and Monographs
Publication Year 2004: Volume 110
ISBNs: 978-0-8218-3515-9 (print); 978-1-4704-1337-8 (online)
MathSciNet review: MR2061425
MSC: Primary 53C44; Secondary 35K60, 53C21
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 book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds.
Graduate students and research mathematicians interested in geometric analysis.
Table of Contents
- 1. The Ricci flow of special geometries
- 2. Special and limit solutions
- 3. Short time existence
- 4. Maximum principles
- 5. The Ricci flow on surfaces
- 6. Three-manifolds of positive Ricci curvature
- 7. Derivative estimates
- 8. Singularities and the limits of their dilations
- 9. Type I singularities
- Appendix A. The Ricci calculus
- Appendix B. Some results in comparison geometry