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