The Ricci flow : an introduction / Bennett Chow, Dan Knopf.

Format:

Book

Author/Creator:

Chow, Bennett, author.

Knopf, Dan, 1959- author.

Series:

Mathematical surveys and monographs ; no. 110.

Mathematical surveys and monographs, 0076-5376 ; volume 110

Language:

English

Subjects (All):

Global differential geometry.

Ricci flow.

Riemannian manifolds.

Physical Description:

1 online resource (342 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2004]

Language Note:

English

Summary:

The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to ``flow'' a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics. Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program. The Ricci flow method is now central to our understanding of the geometry and topology of manifolds. This book is an introduction to that program and to its connection to Thurston's geometrization conjecture. The authors also provide a ``Guide for the hurried reader'', to help readers wishing to develop, as efficiently as possible, a nontechnical appreciation of the Ricci flow program for 3-manifolds, i.e., the so-called ``fast track''. The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds.

Contents:

""Contents""; ""Preface""; ""A guide for the reader""; ""A guide for the hurried reader""; ""Acknowledgments""; ""Chapter 1. The Ricci flow of special geometries""; ""1. Geometrization of three-manifolds""; ""2. Model geometries""; ""3. Classifying three-dimensional maximal model geometries""; ""4. Analyzing the Ricci flow of homogeneous geometries""; ""5. The Ricci flow of a geometry with maximal isotropy SO (3)""; ""6. The Ricci flow of a geometry with isotropy SO (2)""; ""7. The Ricci flow of a geometry with trivial isotropy""; ""Notes and commentary""

""Chapter 2. Special and limit solutions""""1. Generalized fixed points""; ""2. Eternal solutions""; ""3. Ancient solutions""; ""4. Immortal solutions""; ""5. The neckpinch""; ""6. The degenerate neckpinch""; ""Notes and commentary""; ""Chapter 3. Short time existence""; ""1. Variation formulas""; ""2. The linearization of the Ricci tensor and its principal symbol""; ""3. The Ricci�DeTurck flow and its parabolicity""; ""4. The Ricci�DeTurck flow in relation to the harmonic map flow""; ""5. The Ricci flow regarded as a heat equation""; ""Notes and commentary""

""Chapter 4. Maximum principles""""1. Weak maximum principles for scalar equations""; ""2. Weak maximum principles for tensor equations""; ""3. Advanced weak maximum principles for systems""; ""4. Strong maximum principles""; ""Notes and commentary""; ""Chapter 5. The Ricci flow on surfaces""; ""1. The effect of a conformal change of metric""; ""2. Evolution of the curvature""; ""3. How Ricci solitons help us estimate the curvature from above""; ""4. Uniqueness of Ricci solitons""; ""5. Convergence when . . .""; ""6. Convergence when . . .""; ""7. Strategy for the case . . .""

""8. Surface entropy""""9. Uniform upper bounds for . . .""; ""10. Differential Harnack estimates of LYH type""; ""11. Convergence when R(·,0) > 0""; ""12. A lower bound for the injectivity radius""; ""13. The case that R(·,0) changes sign""; ""14. Monotonicity of the isoperimetric constant""; ""15. An alternative strategy for the case χ(M[sup(2)] > 0)""; ""Notes and commentary""; ""Chapter 6. Three-manifolds of positive Ricci curvature""; ""1. The evolution of curvature under the Ricci flow""; ""2. Uhlenbeck's trick""; ""3. The structure of the curvature evolution equation""

""4. Reduction to the associated ODE system""""5. Local pinching estimates""; ""6. The gradient estimate for the scalar curvature""; ""7. Higher derivative estimates and long-time existence""; ""8. Finite-time blowup""; ""9. Properties of the normalized Ricci flow""; ""10. Exponential convergence""; ""Notes and commentary""; ""Chapter 7. Derivative estimates""; ""1. Global estimates and their consequences""; ""2. Proving the global estimates""; ""3. The Compactness Theorem""; ""Notes and commentary""; ""Chapter 8. Singularities and the limits of their dilations""

""1. Classifying maximal solutions""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 317-322) and index.

Description based on print version record.

ISBN:

1-4704-1337-X