Sun-Chin Chu

Abstract: Contents Preface ix What Part II is about ix Highlights and interdependences of Part II xi Acknowledgments xiii Contents of Part II of Volume Two xvii Notation and Symbols xxiii Chapter 10. Weak Maximum Principles for Scalars, Tensors, and Systems 1 1. Weak maximum principles for scalars and symmetric 2-tensors 2 2. Vector bundle formulation of the weak maximum principle for systems 9 3. Spatial maximum function and its Dini derivatives 24 4. Convex sets, support functions, ODEs preserving convex sets 32 5. Proof of the WMP for systems: time-dependent sets and avoidance sets 43 6. Maximum principles for weak solutions of heat equations 47 7. Variants of maximum principles 56 8. Notes and commentary 65 Chapter 11. Closed Manifolds with Positive Curvature 67 1. Multilinear algebra related to the curvature operator 69 2. Algebraic curvature operators and Rm 77 3. A family of linear transformations and their effect on R 2 + R# 89 4. Proof of the main formula for D a ^(R) 94 5. The convex cone of 2-nonnegative algebraic curvature operators 105 6. A pinching family of convex cones in the space of algebraic curvature operators 116 7. Obtaining a generalized pinching set from a pinching family and the proof of Theorem 11.2 126 8. Summary of the proof of the convergence of Ricci flow 134 9. Notes and commentary 136 Chapter 12. Weak and Strong Maximum Principles on Noncompact Manifolds 139 vi CONTENTS

TL;DR: In this article, the Ricci flow was shown to be a solution to the Harnack inequality for a torsion-free connection compatible with a degenerate metric on space-time.