1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

82 2. KAHLER-RICCI FLOW

Moser iteration, and differential Harnack estimates (see the discussion be-
low). The corresponding theory for systems is considerably harder, some-
times tractable only under more restrictive conditions such as the nonnega-
tivity of the bisectional curvature (see Sections 6-8 of this chapter).
Since

(2.53)

we have that

. acp
v(x, t) =;= - at (x, t)

is also a potential function of Ra13(x, t) - ~9a13(x, t). From (2.50), (2.53),
and (2.51), we compute
av a (acp) a/3 a _ r acp
at - -at at - -g at^9 af3 - -:;;, at
r

= /j.v + -v

n
with the initial condition v(x, 0) = f(x, 0). Therefore, if we insist, as in
Lemma 2.42, that the potential function f(x, t) satisfies the heat equation

gtf = /j.f + :!:__ f, we must have

n

(2.54)

Recall from (2.43) that III = l~I ::::; ce!;,t. More precisely, we have the

following.

LEMMA 2.52 (Time-derivative estimate for cp).

(2.55) -C1er> rt ::::; f ( x, t) =-Ft acp ::::; C2er> rt ,

where C1 ~ -minxEMn f(x, 0) and C2 ~ maxxEM f(x, 0).

5.2. Proof of Theorem 2.50. Theorem 2.50 is proved via a progres-

sion of estimates which culminates with a C^2 ,a-estimate for cp (t) on bounded
time intervals. The C^0 -estimate is the following.

LEMMA 2.53 (C^0 -estimate: bound for cp-uniform when ci ::::; 0). If

r i= 0, then

(2.56)

If r = 0, then

-C2t::::; cp(x, t)::::; C 1 t.

PROOF. For the upper bound we compute

cp(x, t) = cp(x, 0) +lat~~ (T) dT::::; lat C1ef;,r dT.