Ricci Flow and the Sphere Theorem

Simon Brendle (Autor)

Buch | Softcover

176 Seiten

2010
American Mathematical Society (Verlag)
978-1-4704-7904-6 (ISBN)

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In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincare conjecture. Furthermore, various convergence theorems have been established. This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H. E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen. This text originated from graduate courses given at ETH Zurich and Stanford University, and it is directed at graduate students and researchers. The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required.

Simon Brendle, Stanford University, CA

Chapter 1. A survey of sphere theorems in geometry
Chapter 2. Hamilton's Ricci flow
Chapter 3. Interior estimates
Chapter 4. Ricci flow on $S^2$
Chapter 5. Pointwise curvature estimates
Chapter 6. Curvature pinching in dimension 3
Chapter 7. Preserved curvature conditions in higher dimensions
Chapter 8. Convergence results in higher dimensions
Chapter 9. Rigidity results
Appendix A. Convergence of evolving metrics
Appendix B. Results from complex linear algebra
Problems

Erscheinungsdatum	19.11.2025
Reihe/Serie	Graduate Studies in Mathematics
Verlagsort	Providence
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-7904-4 / 1470479044
ISBN-13	978-1-4704-7904-6 / 9781470479046
Zustand	Neuware