1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. BASIC RICCI FLOW

PROOF. Let p (t) ~ Rmin (t). Under the unnormalized Ricci flow,

dp > 3._p2 > 0.

dt - n -

Under the normalized Ricci flow,

dp 2

dt 2: -:;;, p (p - r) 2: 0

as long as p :S 0 (note that p - r :S 0 always).

459

D

The weak maximum principle as applied to symmetric 2-tensors says the

following (see Theorem 4.6 on p. 97 of Volume One).^3

THEOREM A.21 (Maximum principle for 2-tensors). Let g ( t) be a smooth

l-parameter family of Riemannian metrics on a closed manifold Mn. Let

a(t) E C^00 (T M ©s T M) be a symmetric (2, 0)-tensor satisfying the semi-

linear heat equation

a

at a 2: ~g(t)C¥ + (3,

where f3 (a, g, t) is a symmetric (2, 0)-tensor which is locally Lipschitz in all

its arguments and satisfies the null eigenvector assumption that

f3 (V, V) (x, t) = (/3ij ViVj) (x, t) 2: 0

at any point and time (x, t) where ( aij WiWj) (x, t) 2: 0 for all W and

( C¥ij vj) (x, t) = 0.

If a (0) 2: 0 {that is, if a (0) is positive semidefinite), then a (t) 2: 0 for all

t 2: 0 such that the solution exists.

Applying this to the evolution equation (Vl-6.10) for the Ricci tensor in
dimension 3 yields the following (Corollary 6.11 on p. 177 of Volume One).
COROLLARY A.22 (3d positive Ricci curvature persists). Let g (t) be a
solution of the Ricci flow on a closed 3-manifold with g (0) = go. If go has
positive {nonnegative) Ricci curvature, then g (t) has positive {nonnegative)
Ricci curvature for as long as the solution exists.
2.3. Uhlenbeck's trick. Uhlenbeck's trick allows us to put the evo-
lution equation (Vl-6.17) satisfied by Rm into a particularly nice form. (See
pp. 180-183 in Section 2 of Chapter 6 in Volume One.) Let (Mn, g (t)),
t E [O, T), be a solution to the Ricci flow with g (0) = go. Let V be a vec-
tor bundle over M isomorphic to TM, and let lo : V --+ TM be a bundle

isomorphism. Then if we define a metric h on V by

(Vl-p. 181a) h ~ lo (go) ,

we automatically obtain a bundle isometry

(Vl-p. 18lb) lo : (V, h)--+ (TM, go).

(^3) The statement we give here is slightly stronger, since in the null eigenvector assump-
tion we also assume that ( IY.ij ViVj) (x, t) ;::::: 0 at (x, t).