The Ricci Flow: Techniques and Applications: Part IV: Long-Time Solutions and Related Topics
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, The College of Charleston, Charleston, SC, Dan Knopf, University of Texas at 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: 2015; Volume 206
ISBNs: 978-0-8218-4991-0 (print); 978-1-4704-2677-4 (online)
MathSciNet review: MR3409114
MSC: Primary 53C44; Secondary 35K55
Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics.
In dimension 3, Perelman completed Hamilton's program to prove Thurston's geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This book discusses recent developments on gradient Ricci solitons, which model the singularities developing under the Ricci flow. In the shrinking case there is a surprising rigidity which suggests the likelihood of a well-developed structure theory. A broader class of solutions is ancient solutions; the authors discuss the beautiful classification in dimension 2. In higher dimensions they consider both ancient and singular Type I solutions, which must have shrinking gradient Ricci soliton models. Next, Hamilton's theory of 3-dimensional nonsingular solutions is presented, following his original work. Historically, this theory initially connected the Ricci flow to the geometrization conjecture. From a dynamical point of view, one is interested in the stability of the Ricci flow. The authors discuss what is known about this basic problem. Finally, they consider the degenerate neckpinch singularity from both the numerical and theoretical perspectives.
This book makes advanced material accessible to researchers and graduate students who are interested in the Ricci flow and geometric evolution equations and who have a knowledge of the fundamentals of the Ricci flow.
Graduate students and researchers interested in geometric evolution equations.
Table of Contents
- Chapter 27. Noncompact gradient Ricci solitons
- Chapter 28. Special ancient solutions
- Chapter 29. Compact 2-dimensional ancient solutions
- Chapter 30. Type I singularities and ancient solutions
- Chapter 31. Hyperbolic geometry and 3-manifolds
- Chapter 32. Nonsingular solutions on closed 3-manifolds
- Chapter 33. Noncompact hyperbolic limits
- Chapter 34. Constant mean curvature surfaces and harmonic maps by IFT
- Chapter 35. Stability of Ricci flow
- Chapter 36. Type II singularities and degenerate neckpinches
- Appendix K. Implicit function theorem