xviii CONTENTS OF PART II OF VOLUME TWO

Refinements of the maximum principle include the case when the sub-

sets with convex fibers are time-dependent and also when there is a so-called

avoidance set for the solutions of the PDE. In terms of applications, one of

the most important special cases is when the sections of the bundle are bi-

linear forms. We discuss this case and applications to the curvature tensors.

We also discuss the Aleksandrov-Bakelman-Pucci maximum principle for

elliptic equations.

Chapter 11. In this chapter we present Bohm and Wilking's solu-

tion to the conjecture of Rauch and Hamilton on the classification of closed

Riemannian manifolds with positive curvature

operator.1

The flavor of this

chapter is more algebraic with an essential component of the proof being the

irreducible decomposition of (algebraic) curvature tensors. Generally, one

of the ideas is to study when linear transformations of convex preserved sets

(with respect to the ODE corresponding to the PDE satisfied by Rm) remain

preserved. More specifically, via a 1-parameter family of linear transforma-

tions the cone of 2-nonnegative curvature operators is mapped into the cone

of nonnegative curvature operators. This reduces the classification prob-

lem for Riemannian manifolds with 2-positive curvature operator to that for

manifolds with positive curvature operator. To study the latter problem,

one would hope that the cones of 2-positive curvature operators with arbi-

trary Ricci pinching are preserved. Unfortunately there does not seem to

be any known way to prove this. Instead, one can prove that suitable linear

transformations of the cones of 2-positive curvature operators with arbitrary

Ricci pinching are preserved. This is sufficient to prove the Rauch-Hamilton

conjecture.

Chapter 12. This chapter comprises two main topics: (1) weak max-

imum principles on noncompact manifolds and (2) the strong maximum

principle, which is a local result. In both cases we consider scalar parabolic

equations and systems of parabolic equations.

Since singularity models are often noncompact, it is important to be

able to apply the weak maximum principle on complete, noncompact man-

ifolds. We begin with the heat equation and present the weak maximum

principle of Karp and Li which applies to solutions with growth slower than

exponential quadratic in distance. Using barrier functions, we then give a

weak maximum principle for bounded solutions of heat-type systems. As a

special case, we show that complete solutions to the Ricci flow on noncom-

pact manifolds, with nonnegative curvature operator initially and bounded

curvature on space and time, have nonnegative curvature operator for all

time.

We also discuss mollifiers on Riemannian manifolds with a lower bound

on the injectivity radius. One technical issue we discuss, using the mollifiers

obtained above, is the construction of distance-like functions with bounds on

They actually obtain the stronger result of classifying closed manifolds with 2-

positive curvature operator.