X l l

PREFACE

(6) We present Perelman's differential Harnack estimate for fundamen-

tal solutions of the adjoint heat equation coupled to the Ricci flow.

This result, of which we give a detailed proof, will be used in the

proof of Perelman's pseudolocality discussed in Part III.

(7) We review tensor calculus on the frame bundle—a framework for

proving Hamilton's matrix Harnack estimate for the Ricci flow.

The appendices are intended to make this book more self-contained.

0.2. Interdependencies. The chapters are for the most part indepen-

dent. However, many of the results discussed in this book rely on various

forms of the maximum

principle.5

For example we have the following re-

liances.

(1) Chapter 11 on manifolds with positive curvature operator, Section

1 of Chapter 13 on curvature conditions that are not preserved,

and Chapter 15 on the matrix Harnack estimate all require the

(time-independent) maximum principles for tensors and systems.

(2) The maximum principles on noncompact manifolds in Chapter 12

require some familiarity with the maximum principles on compact

manifolds in Chapter 10.

(3) Chapter 14 on local derivative estimates, Section 2 of Chapter 13

on the real analyticity of solutions, and Chapter 16 on Perelman's

differential Harnack estimate all require the maximum principle for

scalars.

5Perhaps we may say, 'Geometric analysis is simple; just apply the maximum

principle!'