1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

458 A. BASIC RICCI FLOW THEORY

THEOREM A.17 (Short-time existence on noncompact manifolds). Let

M be a noncompact manifold and let go be a complete metric with bounded

sectional curvature. There exists a complete solution g (t), t E [O, T), of the

Ricci flow with g (0) =go and curvature bounded on compact time intervals.

This solution is unique in the class of complete solutions with curvature

bounded on compact time intervals.

2.2. Maximum principles for scalars, tensors and systems. A

form of the scalar maximum principle useful for the Ricci flow is the following
(Theorem 4.4 on p. 96 of Volume One).

THEOREM A.18 (Scalar maximum principle: ODE to PDE). Let u: Mn x

[O, T) -> IR be a C^2 function on a closed manifold satisfying

au

at :S ~g(t)U + (X, Vu)+ F (u)

and u (x, 0) :::;; C for all x E M, where g (t) is a 1-parameter family of

metrics and F is locally Lipschitz. Let <p (t) be the solution to the initial-

value problem

d<p

dt = F (1.p)'

<p (0) = C.

Then

u (x, t) :::;; <p(t)

for all x EM and t E [O, T) such that <p (t) exists.

Since under the Ricci flow, by (Vl-6.6) ~~ ~ ~R + ~R^2 , we have the

following (see the proof of Lemma 6.53 on pp. 209-210 of Volume One).

COROLLARY A.19 (Scalar curvature lower bound). Let (Mn, g ( t)), where

0 :::;; t < T, be a solution of the Ricci flow for which the maximum principle

holds. If infxEMn R (x, to)~ p > 0 for some to E [O, T), then

Rinf (t) ~ inf R (x, t) ~ 1 2 ~.

xEM p - n t - to)

In particular, g (t) becomes singular in finite time.

Moreover, we have the following.

LEMMA A.20 (Minimum scalar curvature monotonicity).

(1) Under the unnormalized Ricci flow the minimum scalar curvature

is a nondecreasing function of time.

(2) Under the normalized Ricci flow the minimum scalar curvature is

nondecreasing as long as it is nonpositive.