The Ricci flow in Riemannian geometry: a complete proof of the differentiable 1/4-pinching sphere theorem

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 andBrendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem. A self contained presentation of the proof of the differentiable sphere theorem A presentation of the geometry of vector bundles in a form suitable for geometric PDE A discussion of the history of the sphere theorem and of future challenges INDICE: 1 Introduction.- 2 Background Material.- 3 Harmonic Mappings.- 4 Evolution of the Curvature.- 5 Short-Time Existence.- 6 Uhlenbeck’s Trick.- 7 The Weak Maximum Principle.- 8 Regularity and Long-Time Existence.- 9 The Compactness Theorem for Riemannian Manifolds.- 10 The F-Functional and Gradient Flows.- 11 The W-Functional and Local Noncollapsing.- 12 An Algebraic Identity for Curvature Operators.- 13 The Cone Construction of Böhm and Wilking.- 14 Preserving Positive Isotropic Curvature.- 15 The Final Argument

• ISBN: 978-3-642-16285-5
• Editorial: Springer
• Encuadernacion: Rústica
• Páginas: 276
• Fecha Publicación: 01/12/2010
• Nº Volúmenes: 1
• Idioma: Inglés