The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem (Lecture Notes in Mathematics, Vol. 2011) (Lecture Notes in Mathematics (2011))

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. 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 Introduction -- Background Material -- Harmonic Mappings -- Evolution Of The Curvature -- Short-time Existence -- Uhlenbeck's Trick -- The Weak Maximum Principle -- Regularity And Long-time Existence -- The Compactness Theorem For Riemannian Manifolds -- The F-functional And Gradient Flows -- The W-functional And Local Noncollapsing -- An Algebraic Identity For Curvature Operators -- The Cone Construction Of Böhm And Wilking -- Preserving Positive Isotropic Curvature -- The Final Argument. Ben Andrews, Christopher Hopper. Includes Bibliographical References (p. 287-292) And Index. Annotation 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