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Ricci Flow and the Sphere Theorem

Simon Brendle Stanford University, Stanford, CA
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Hardcover ISBN: 978-0-8218-4938-5
Product Code: GSM/111
List Price: $53.00 MAA Member Price:$47.70
AMS Member Price: $42.40 Electronic ISBN: 978-1-4704-1173-2 Product Code: GSM/111.E List Price:$50.00
MAA Member Price: $45.00 AMS Member Price:$40.00
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List Price: $79.50 MAA Member Price:$71.55
AMS Member Price: $63.60 Click above image for expanded view Ricci Flow and the Sphere Theorem Simon Brendle Stanford University, Stanford, CA Available Formats:  Hardcover ISBN: 978-0-8218-4938-5 Product Code: GSM/111  List Price:$53.00 MAA Member Price: $47.70 AMS Member Price:$42.40
 Electronic ISBN: 978-1-4704-1173-2 Product Code: GSM/111.E
 List Price: $50.00 MAA Member Price:$45.00 AMS Member Price: $40.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$79.50 MAA Member Price: $71.55 AMS Member Price:$63.60
• Book Details

Volume: 1112010; 176 pp
MSC: Primary 53;

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.

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

• Chapters
• 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

• Reviews

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1112010; 176 pp
MSC: Primary 53;

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.

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

• Chapters
• 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
• 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
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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