Contents

Preface ix

What Part II is about ix

Highlights and interdependencies 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 Dajb(H) 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