CHAPTER 10

Weak Maximum Principles for Scalars, Tensors,

and Systems

And if you go chasing rabbits ...

— From "White Rabbit" by Grace Slick of Jefferson Airplane

In this chapter we present (starting in Section 2) the statements and

proofs of various versions of the weak maximum principle for systems. We

restrict ourselves to closed manifolds and in Chapter 12 we discuss the com-

plete noncompact case. The preliminary formulation we give is for sections

of vector bundles solving heat-type equations where the Riemannian metrics

on the base manifold and the connections on the vector bundle are time-

dependent (for example, the metrics solve the Ricci flow equation) whereas

the metric on the fibers is fixed. In the study of the Ricci flow we are able

to reduce to the fixed fiber-metric case by Uhlenbeck's trick (see Section 2

of Chapter 6 in Volume One). In a sense, all proofs of maximum principles

for classical solutions boil down to the second derivative test in calculus.

The main application in the Ricci flow has been to the study of the evolu-

tion of the Riemann curvature

tensor.1

Another application is to the proof

of Hamilton's matrix Harnack estimate for the Ricci flow (see Section 4 of

Chapter 15).

We first review the statements and some applications of the weak max-

imum principles for scalars and tensors which were discussed in Volume

One. Then we give proofs of the Weinberger-Hamilton maximum principle

for systems including some generalized versions, e.g., time-dependent fiber-

wise convex sets and the avoidance maximum principle. Regarding the case

where the fiberwise convex sets depend continuously on time, in subsection

3.4 we give Bohm and Wilking's simplified proof. As applications of the

aforementioned maximum principles, in this chapter we briefly survey the

following results, most of which were proved in Volume One and are due to

Hamilton.2

Applications of the scalar maximum principles are

(1) lower bound for the scalar curvature,

Many of the maximum principles in this chapter also apply to other geometric flows

such as the mean curvature flow.

Independently by Ivey in the case of the Hamilton-Ivey estimate. For the classific-

cation of 4-manifolds with positive isotropic curvature there is work of Chen-Zhu [110]

on fixing a gap in Hamilton's proof partially using Perelman's work.

1

http://dx.doi.org/10.1090/surv/144/01