(c) (in real dimension 2) exponential decay of
\2VVf-(Af)g\ = \(R-r)g-£Vfg\,
which measures the difference from being a gradient soliton, when R c 0
(see Corollary 5.35 in Volume One for this last estimate),
(3) (dimensions 2 and 3) isoperimetric estimates (for dimension 2, see
Section 14 of Chapter 5 in Volume One, and for dimension 3, see §7 of
Chapter 5 of [146]).
1.2. Maximum principle for 2-tensors. Since we are interested in
the evolution of the Riemann and Ricci curvature tensors under the Ricci
flow, we have previously considered the maximum principle for tensors (see
Chapter 4 of Volume One). Now we recall a special form of the weak max-
imum principle for symmetric 2-tensors satisfying heat-type equations con-
cerning when these 2-tensors remain nonnegative definite. For linear para-
bolic systems on Euclidean space this is due to Stys [460]; the general case of
symmetric 2-tensors on evolving Riemannian manifolds is due to Hamilton
[244]. First we make the following definition.
10.6 (Null eigenvector assumption). Let
(3: (TM*®STM*) x [ 0 , T ) ^ T M * % T M * .
We say that (3 satisfies the null-eigenvector assumption if the following
condition holds. If a nonnegative symmetric 2-tensor uo 0 at a point x e M
and a vector V G TMX are such that
0, then
0, for any t G [0,T).
The following is a slightly stronger version of Theorem 4.6 on p. 97 in
Volume One; its proof is exactly the same. This result also holds for com-
pact manifolds with boundary as long as we have a (x, t) 0 on the side
10.7 (Maximum principle for 2-tensors). Let
t G
[0,T), be a closed manifold with a family of Riemannian metrics smoothly
depending on time and let a{t) G C°° (TM* ®s TM*), t G [0,T), be a
symmetric 2-tensor satisfying
(10.16) Ag(t)a + (X(t),Va) + (3,
where X (t) is a time-dependent vector field and (3 f3 (a, t) is a symmetric
2-tensor, which is a locally Lipschitz function of a and is continuous in t.
Suppose that (3 satisfies the null-eigenvector assumption. If a(0) 0; then
a(t) 0 for allte [0,T).
Theorem 10.7 has been applied to the Ricci flow on 3-dimensional closed
manifolds to prove the following.
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