2 10. WEAK MAXIMUM PRINCIPLES FOR SCALARS, TENSORS, AND SYSTEMS

(2) trace Harnack estimate on surfaces,

(3) doubling-time estimate for |Rm|.

Applications of the maximum principle for tensors are

(1) in dimension 3,

(a) nonnegative Ricci curvature is preserved,

(b) nonnegative sectional curvature is preserved,

(c) Ricci pinching is preserved,

(2) matrix Harnack estimate on surfaces.

Applications of the maximum principle for systems are

(1) in dimension 3,

(a) Ricci pinching improves,

(b) Hamilton-Ivey estimate,

(2) in dimension 4 (these results were not discussed in Volume One),

(a) classification of manifolds with positive curvature operator,

(b) classification of manifolds with positive isotropic curvature (with

no essential incompressible 3-dimensional spherical space forms).

Additional applications of maximum principles occur in the study of

the Kahler-Ricci flow (see Chapter 2 of Part I of this volume) and the

linearized Ricci flow (see Chapter 20 of Part III of this volume). Volume One

discussed some applications to the Ricci flow on surfaces, where the estimates

are motivated by the consideration of quantities which either vanish or are

constant on gradient Ricci solitons. The reader who finds Section 1 too

repetitive may skip directly to Section 2.

In Section 6 we discuss maximum principles for weak solutions of heat

equations in the form of weighted energy estimates. In Section 7 we consider

the Aleksandrov-Bakelman-Pucci maximum principle which provides a C°-

estimate for linear elliptic and Monge-Ampere equations. The method of

its proof is geometric and involves taking the concave envelope of a function

and the fact that the Jacobian of the gradient map of a function is the

absolute value of the determinant of the Hessian of the function.

As throughout the book, unless otherwise indicated, all objects con-

sidered in this chapter will be smooth and all manifolds will be oriented;

however, most arguments shall hold for

C2,1

functions or sections of vector

bundles, i.e., those functions or sections having continuous space derivatives

of second order and continuous time derivative (of first order).

1. Weak maximum principles for scalars and symmetric 2-tensors

In this section we review the weak maximum principles (WMP)

for scalars and tensors, which have been applied to obtain curvature and

Harnack estimates. The purpose of this discussion is to collect in one place

some scattered results proved using maximum principles.

1.1. Scalar maximum principle. Let

Mn

be a closed manifold and

let g (£), t G [0, T), be a family of metrics on M. The Laplacian is defined

by Ag(t) =F g

(t)lJ

V (t)i V (t)., where V (t) denotes the covariant derivative