Preface Chapter 1 Introduction 1.1 A Quick Tour of Three-Manifolds 1.2 Background Material in Riemannian Geometry Chapter 2 Existence and Uniqueness of Ricci Flow 2.1 Examples of Ricci Flow 2.2 Variational Formulas for Tensors 2.3 Ricci-DeTurck Flow 2.4 Uniqueness of Ricci Flow Chapter 3 Maximum Principle 3.1 Maximum Principle for Functions 3.2 Maximum Principle for Parabolic Systems 3.3 Maximum Principle on Noncompact Manifolds 3.4 Strong Maximum Principle I 3.5 Strong Maximum Principle II Chapter 4 Curvature Operator 4.1 Evolution Equation for the Curvature Operator 4.2 Shi's Higher-Order Estimate 4.3 Extension of Ricci Flow 4.4 Maximum Principles for the Curvature Operator 4.5 Uhlenbeck's Trick 4.6 Curvature Operator and Holonomy Group Chapter 5 Hamilton's ODE and Its Applications 5.1 Convex Sets in Euclidean Space 5.2 Curvature Operator and Hamilton's ODE 5.3 Hamilton-Ivey Pinching Estimate 5.4 Positive Curvature Conditions Preserved by Ricci Flow 5.5 Nonnegative Isotropic Curvature Conditions 5.6 Nonnegative Bisectional Curvature Chapter 6 Ricci Flow and Sphere Theorem 6.1 Pinching Towards Constant Curvature 6.2 Proof of the Sphere Theorem 6.3 Construction of Cones Chapter 7 Entropy and Monotonicity on Ricci Flow 7.1 Basics for the Heat Kernel 7.2 Perelman's Entropy and No-Local-Collapsing 7.3 Monotonicity along the Ricci Flow 7.4 Poincare Inequality and Logarithmic Sobolev Inequality Chapter 8 Heat Kernel Estimates with Bounded Entropy 8.1 Ultracontractivity of the Heat Equation 8.2 Center of the Conjugate Heat Measure 8.3 Pointwise Upper Bound of the Heat Kernel 8.4 Differential Harnack Inequality 8.5 Pointed Entropy of the Ricci Flow Chapter 9 Structure of Ricci Shrinkers 9.1 Basic Properties of Ricci Shrinkers 9.2 Classification of 2-dimensional Ricci Shrinkers 9.3 Classification of 3-dimensional Ricci Shrinkers 9.4 Classification of Ricci Shrinkers with Nonnegative Curvature 9.5 Entropy of the Ricci Shrinker
Chapter 10 Convergence of Ricci Flows 10.1 Metric Spaces and Gromov-Hausdorff Convergence 10.2 Convergence of Riemannian Manifolds 10.3 Hamilton's Compactness Theorem 10.4 Noncollapsed Ricci Flow Limit Spaces Chapter 11 Analysis of Ancient Solutions 11.1 Matrix Harnack Inequality 11.2 Asymptotic Scalar Curvature and Volume Ratio 11.3 Compactness of Ancient Solutions 11.4 Asymptotic Ricci Shrinker 11.5 Ancient Solutions in Low Dimensions Chapter 12 Ricci Flow with Surgery 12.1 Canonical Neighborhoods Theorem 12.2 Ricci Flow at the First Singular Time 12.3 Cutoff Parameters and (r, δ)-Surgery 12.4 Finite-Time Extinction 12.5 Thick-Thin Decomposition Appendix I Quasi-Linear Parabolic Equations II Entropy of Product Riemannian Manifolds III Ricci Flow of Type-I Bibliography Index
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