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

(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.

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

uJijV1

— 0, then

(10.15)

/3ij(uJt)ViVj

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

boundary.

THEOREM

10.7 (Maximum principle for 2-tensors). Let

(Mn,g(t)),

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

dot

(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.