| Peter Topping is a Senior Lecturer in Mathematics at the University of Warwick. |
| 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 1.4 The topology and geometry of manifolds in low dimensions 1.5 Using Ricci flow to prove topological and geometric results 2 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 4.3 Generalisation to Vector Bundles 4.4 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 5-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-4A + R 7 Compactness of Riemannian manifolds and flows 7.1 Convergence and compactness of manifolds 7.2 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 bundles 9.6 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 A Connected sum References Index |
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