Ricci Flow and the Sphere Theorem: 111 (Graduate Studies in Mathematics)

Brendle, Simon

ISBN 10: 1470479044 / ISBN 13: 9781470479046

Published by American Mathematical Society, 2010

About this Item

2010. paperback. . . . . .

Seller Inventory # V9781470479046

Bibliographic Details

• Title: Ricci Flow and the Sphere Theorem: 111 (Graduate Studies in Mathematics)
• Publisher: American Mathematical Society
• Publication Date: 2010
• Language: English
• ISBN 10: 1470479044
• ISBN 13: 9781470479046
• Binding: Soft cover
• Condition: New

About this title

Synopsis

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 Poincaré 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 Zürich 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.