1547671870-The_Ricci_Flow__Chow

228 7. DERIVATIVE ESTIMATES

when we differentiate f\7 Rmf^2 by the good term -2(3 f\7 Rmf^2 we get when
we differentiate fRmf^2. Moreover, we get an upper bound F :S (3K^2 at

t = 0 by hypothesis on fRmf, without needing further assumptions on the

initial metric. Combining equation (7.2) with the result of Lemma 7.4 and

discarding the term - 2t IV^2 Rml^2 , we observe that F satisfies the differential
inequality

a

at F :S b..F + (1 +cit fRmf - 2(3) f\7 Rmf^2 + c2f3 fRmf^3 ,

where the constants c1, c2 depend only on n. By hypothesis, fRmf :SK for

all t E [O, a/ K]. So for such times,

a 2 3

ot F :S b..F + (1 + C1a - 2(3) f\7 Rml + c2f3K.

Choose (3 2: ( 1 + c1 a) / 2, noting that (3 depends only on n and max {a, 1}.

Then for t E [O, a/ K],

a

at F :S b..F + c2(3K^3.

Thus by the parabolic maximum principle, we have

sup F (x, t) :S (3K^2 + c2f3K^3 t :S (1 + c2a) (3K^2 :S Cf K^2

xEMn

for 0 :S t :S a/ K, where C 1 is a constant depending only on n and max {a, 1}.

Hence

a

for 0 < t :S K.

This completes the proof of the case m = 1.

Now by induction, we may assume that we have estimated IVj Rml for

all 1 :S j < m. Let 1 :S k :S m. We begin by computing

(7.3) :t lvkRml

2

= 2(%t (vkRm) ,'VkR) +Re (vkRm)

2

.

To evaluate the time derivative on the right-hai:.d side, we again recall for-

mulas (6.1) and (6.17) and write

k

= \lk (t:::..Rm+(Rm)*^2 ) + L\7jRm*\7k-jRm

j=l

k

= \lk b.. Rm+ L \lj Rm* vk-j Rm.

j=O