**Graduate Studies in Mathematics**

Volume: 111;
2010;
176 pp;
Hardcover

MSC: Primary 53;

Print ISBN: 978-0-8218-4938-5

Product Code: GSM/111

List Price: $50.00

Individual Member Price: $40.00

**Electronic ISBN: 978-1-4704-1173-2
Product Code: GSM/111.E**

List Price: $50.00

Individual Member Price: $40.00

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#### Supplemental Materials

# Ricci Flow and the Sphere Theorem

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*Simon Brendle*

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.

#### Readership

Graduate students and research mathematicians interested in differential geometry and topology of manifolds.

#### Reviews & Endorsements

This book is a great self-contained presentation of one of the most important and exciting developments in differential geometry. It is highly recommended for both researchers and students interested in differential geometry, topology and Ricci flow. As the main technical tool used in the book is the maximum principle, familiar to any undergraduate, this book would make a fine reading course for advanced undergraduates or postgraduates and, in particular, is an excellent introduction to some of the analysis required to study Ricci flow.

-- Huy The Nyugen, Bulletin of the LMS

This is an excellent self-contained account of exciting new developments in mathematics suitable for both researchers and students interested in differential geometry and topology and in some of the analytic techniques used in Ricci flow. I very strongly recommend it.

-- Jahresbericht Der Deutschen Mathematiker - Vereinigung

#### Table of Contents

# Table of Contents

## Ricci Flow and the Sphere Theorem

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface v6 free
- A survey of sphere theorems in geometry 110 free
- Hamilton’s Ricci flow 1524
- Interior estimates 3140
- Ricci flow on 𝑆² 3746
- Pointwise curvature estimates 4958
- Curvature pinching in dimension 3 6776
- Preserved curvature conditions in higher dimensions 7382
- Convergence results in higher dimensions 101110
- Rigidity results 121130
- Convergence of evolving metrics 155164
- Results from complex linear algebra 159168
- Problems 163172
- Bibliography 169178
- Index 175184 free
- Back Cover Back Cover1186