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The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem

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This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Author(s): Ben Andrews, Christopher Hopper (auth.)
Series: Lecture Notes in Mathematics 2011
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2011

Language: English
Pages: 302
Tags: Partial Differential Equations; Differential Geometry; Global Analysis and Analysis on Manifolds

Front Matter....Pages i-xvii
Introduction....Pages 1-9
Background Material....Pages 11-47
Harmonic Mappings....Pages 49-62
Evolution of the Curvature....Pages 63-82
Short-Time Existence....Pages 83-95
Uhlenbeck’s Trick....Pages 97-113
The Weak Maximum Principle....Pages 115-135
Regularity and Long-Time Existence....Pages 137-143
The Compactness Theorem for Riemannian Manifolds....Pages 145-159
The $$\mathcal{F}$$ -Functional and Gradient Flows....Pages 161-171
The $$\mathcal{W}$$ -Functional and Local Noncollapsing....Pages 173-191
An Algebraic Identity for Curvature Operators....Pages 193-221
The Cone Construction of Böhm and Wilking....Pages 223-233
Preserving Positive Isotropic Curvature....Pages 235-258
The Final Argument....Pages 259-269
Back Matter....Pages 287-296