Lectures on the Ricci Flow
Ricci flow e-böcker
Cambridge University Press
2006
EISBN 9780511721465
Cover; Title; Copyright; Contents; Preface; 1 Introduction; 1.1 Ricci flow: what is it, and from where did it come?; 1.2 Examples and special solutions; 1.2.1 Einstein manifolds; 1.2.2 Ricci solitons; 1.2.3 Parabolic rescaling of Ricci flows; 1.3 Getting a feel for Ricci flow; 1.3.1 Two dimensions; 1.3.2 Three dimensions; The neck pinch; The degenerate neck pinch; Infinite time behaviour; 1.4 The topology and geometry of manifolds in low dimensions; Dimension 1; Dimension 2; Dimension 3; Dimension 4; 1.5 Using Ricci flow to prove topological and geometric results; Dimension 2; Dimension 3.
4.3 Generalisation to Vector Bundles4.4 Properties of parabolic equations; 5 Existence theory for the Ricci flow; 5.1 Ricci flow is not parabolic; 5.2 Short-time existence and uniqueness: The DeTurck trick; 5.3 Curvature blow-up at finite-time singularities; 6 Ricci flow as a gradient flow; 6.1 Gradient of total scalar curvature and related functionals; 6.2 The F -functional; 6.3 The heat operator and its conjugate; 6.4 A gradient flow formulation; 6.5 The classical entropy; 6.6 The zeroth eigenvalue of 4+R; 7 Compactness of Riemannian manifolds and flows.
7.1 Convergence and compactness of manifolds7.2 Convergence and compactness of flows; 7.3 Blowing up at singularities I; 8 Perelman's W entropy functional; 8.1 Definition, motivation and basic properties; 8.2 Monotonicity of W; 8.3 No local volume collapse where curvature is controlled; 8.4 Volume ratio bounds imply injectivity radius bounds; 8.5 Blowing up at singularities II; 9 Curvature pinching and preserved curvature properties under Ricci flow; 9.1 Overview; 9.2 The Einstein Tensor, E; 9.3 Evolution of E under the Ricci flow; 9.4 The Uhlenbeck Trick.
9.5 Formulae for parallel functions on vector bundles9.6 An ODE-PDE theorem; 9.7 Applications of the ODE-PDE theorem; 10 Three-manifolds with positive Ricci curvature, and beyond; 10.1 Hamilton's theorem; 10.2 Beyond the case of positive Ricci curvature; Appendix A Connected sum; References; Index.
Dimension 42 Riemannian geometry background; 2.1 Notation and conventions; 2.2 Einstein metrics; 2.3 Deformation of geometric quantities as the Riemannian metric is deformed; 2.3.1 The formulae; 2.3.2 The calculations; 2.4 Laplacian of the curvature tensor; 2.5 Evolution of curvature and geometric quantities under Ricci flow; 3 The maximum principle; 3.1 Statement of the maximum principle; 3.2 Basic control on the evolution of curvature; 3.3 Global curvature derivative estimates; 4 Comments on existence theory for parabolic PDE; 4.1 Linear scalar PDE; 4.2 The principal symbol.
An introduction to Ricci flow suitable for graduate students and research mathematicians.
4.3 Generalisation to Vector Bundles4.4 Properties of parabolic equations; 5 Existence theory for the Ricci flow; 5.1 Ricci flow is not parabolic; 5.2 Short-time existence and uniqueness: The DeTurck trick; 5.3 Curvature blow-up at finite-time singularities; 6 Ricci flow as a gradient flow; 6.1 Gradient of total scalar curvature and related functionals; 6.2 The F -functional; 6.3 The heat operator and its conjugate; 6.4 A gradient flow formulation; 6.5 The classical entropy; 6.6 The zeroth eigenvalue of 4+R; 7 Compactness of Riemannian manifolds and flows.
7.1 Convergence and compactness of manifolds7.2 Convergence and compactness of flows; 7.3 Blowing up at singularities I; 8 Perelman's W entropy functional; 8.1 Definition, motivation and basic properties; 8.2 Monotonicity of W; 8.3 No local volume collapse where curvature is controlled; 8.4 Volume ratio bounds imply injectivity radius bounds; 8.5 Blowing up at singularities II; 9 Curvature pinching and preserved curvature properties under Ricci flow; 9.1 Overview; 9.2 The Einstein Tensor, E; 9.3 Evolution of E under the Ricci flow; 9.4 The Uhlenbeck Trick.
9.5 Formulae for parallel functions on vector bundles9.6 An ODE-PDE theorem; 9.7 Applications of the ODE-PDE theorem; 10 Three-manifolds with positive Ricci curvature, and beyond; 10.1 Hamilton's theorem; 10.2 Beyond the case of positive Ricci curvature; Appendix A Connected sum; References; Index.
Dimension 42 Riemannian geometry background; 2.1 Notation and conventions; 2.2 Einstein metrics; 2.3 Deformation of geometric quantities as the Riemannian metric is deformed; 2.3.1 The formulae; 2.3.2 The calculations; 2.4 Laplacian of the curvature tensor; 2.5 Evolution of curvature and geometric quantities under Ricci flow; 3 The maximum principle; 3.1 Statement of the maximum principle; 3.2 Basic control on the evolution of curvature; 3.3 Global curvature derivative estimates; 4 Comments on existence theory for parabolic PDE; 4.1 Linear scalar PDE; 4.2 The principal symbol.
An introduction to Ricci flow suitable for graduate students and research mathematicians.
