CONTENTS OF PART II OF VOLUME TWO

xix

their gradients and upper bounds of their Hessians on complete Riemannian

manifolds with bounded curvature. These distance-like functions are used

to construct the barrier functions referred to in the previous paragraph.

Moreover, this construction carries over to the case of the Ricci flow.

The aforementioned mollifiers are not only applicable to the proof of the

compactness theorem in regards to the center of mass (discussed in Chap-

ter 4 of Part I of this volume), but also to constructing a barrier function

used in the proof of Hamilton's matrix Harnack estimate in Chapter 15

on complete noncompact Riemannian manifolds with bounded nonnegative

curvature operator.

A fundamental property of the heat equation is that a solution which is

initially nonnegative immediately becomes either positive or identically zero.

This property is known as the strong maximum principle. For a solution

to the Ricci flow, the curvature tensor satisfies a heat-type equation. The

strong maximum principle for systems, due to Hamilton based on earlier

work of Weinberger and others, tells us that for solutions to the Ricci flow

with nonnegative curvature operator (such as singularity models in dimen-

sion 3) the curvature operator has a special form after the initial time. In

particular, the image of the curvature operator is independent of time and

invariant under parallel translation in space. Moreover, using the natural

Lie algebra structure on the fibers A^, x G M, of the bundle of 2-forms, the

image of the curvature operator is a Lie subalgebra. It is useful to observe

that the strong maximum principle is a local result and does not require the

solution to be complete or to have bounded curvature. We also formulate

the strong maximum principle in the more general setting of Chapter 10,

i.e., for sections of a vector bundle which solve a PDE.

In dimension 3, A^ is isomorphic to 50 (3) and its only proper nontrivial

Lie subalgebras are isomorphic to 50 (2), which is 1-dimensional. Hence,

after the initial time, a solution to the Ricci flow on a 3-manifold with non-

negative sectional curvature either is flat, has positive sectional curvature,

or admits a global parallel 2-form. In the last case, by taking the dual of this

2-form, we have a parallel 1-form and the solution splits locally as the prod-

uct of a surface solution and a line. This classification is useful in studying

the singularities which arise in dimension 3.

Chapter 13. In this chapter we discuss the following two topics in Ricci

flow.

(1) Curvature conditions that are not preserved. The weak positivity

(i.e., nonnegativity) of the following curvatures are preserved: scalar curva-

ture, Ricci curvature in dimension 3, Riemann curvature operator, isotropic

curvature, complex sectional curvature, as well as the 2-nonnegativity of

the curvature operator. On the other hand, in this chapter we discuss some

nonnegativity conditions which are not preserved under the Ricci flow. In

particular, we consider the conditions of nonnegative Ricci curvature and

nonnegative sectional curvature in dimensions at least 4 for solutions of the

Ricci flow on both closed and noncompact manifolds.