CONTENTS OF PART II OF VOLUME TWO

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(gradient) Ricci solitons as discussed in Chapter 1 of Part I of this vol-

ume, provide useful bounds for the solution. The general form of the main

estimate, which holds for complete solutions with bounded nonnegative cur-

vature operator and which is known as Hamilton's matrix Harnack estimate,

says that a certain tensor involving two and fewer derivatives of the curva-

ture is nonnegative definite.

The trace Harnack estimate, as the name suggests, is obtained by trac-

ing the matrix inequality. This trace inequality has several important conse-

quences, including the fact that scalar curvature does not decrease too fast

in the sense that for a fixed point, t times the scalar curvature is a nonde-

creasing function of time. More generally, the trace inequality yields a lower

bound for the scalar curvature at a point and time, in terms of the scalar

curvature at any other point and earlier time, and the distance between the

points and the time difference.

We begin the proof of the Harnack estimate with the case of surfaces

(dimension 2), in which case the evolution equation for the Harnack qua-

dratic simplifies (in comparison to higher dimensions) and one can prove the

trace inequality directly. This is unlike the situation in higher dimensions,

where the trace inequality apparently can only be demonstrated by proving

the matrix inequality and then tracing.

In all dimensions, we recall the terms in the matrix Harnack quadratic

obtained in Chapter 1 (p. 9) of Part I of this volume by differentiating

the expanding gradient Ricci soliton equation. The Harnack calculations

simplify when one uses the formalism, given in Appendix F, of considering

tensors as vector-valued functions on the frame bundle. Using the above

formalism, we present the evolution of Harnack calculations which are long

but relatively straightforward. The evolution of the Harnack quantity looks

formally similar to the evolution of the Riemann curvature operator and as

such is amenable to the application of the weak maximum principle when

the solution has nonnegative curvature operator. When the manifold is

noncompact, the techniques used to enable this application are reminiscent

of the techniques used to prove the maximum principle for functions.

We also give a variant on Hamilton's proof of the matrix Harnack esti-

mate, based on reducing the problem to showing that a symmetric 2-tensor

is nonnegative definite.

Chapter 16. In this chapter we give a proof of Perelman's differential

Harnack-type inequality for solutions of the adjoint heat equation coupled

to the Ricci flow. We begin by considering entropy and differential Har-

nack estimates for the heat equation. Our approach for proving Perelman's

differential Harnack-type inequality is to first prove gradient estimates for

positive solutions of the (adjoint) heat equation. Using this and heat kernel

estimates, we give a proof that for solutions to the adjoint heat equation

coupled to the Ricci flow, Perelman's Harnack quantity (or, geometrically,