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Ricci Flow and the Sphere Theorem
 
Simon Brendle Stanford University, Stanford, CA
Ricci Flow and the Sphere Theorem
Hardcover ISBN:  978-0-8218-4938-5
Product Code:  GSM/111
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-1173-2
Product Code:  GSM/111.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-4938-5
eBook: ISBN:  978-1-4704-1173-2
Product Code:  GSM/111.B
List Price: $184.00 $141.50
MAA Member Price: $165.60 $127.35
AMS Member Price: $147.20 $113.20
Ricci Flow and the Sphere Theorem
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Ricci Flow and the Sphere Theorem
Simon Brendle Stanford University, Stanford, CA
Hardcover ISBN:  978-0-8218-4938-5
Product Code:  GSM/111
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-1173-2
Product Code:  GSM/111.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-4938-5
eBook ISBN:  978-1-4704-1173-2
Product Code:  GSM/111.B
List Price: $184.00 $141.50
MAA Member Price: $165.60 $127.35
AMS Member Price: $147.20 $113.20
  • Book Details
     
     
    Graduate Studies in Mathematics
    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.

    Readership

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

  • Table of Contents
     
     
    • 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 publishers of book reviews
    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.

Readership

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 publishers of book reviews
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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