1547671870-The_Ricci_Flow__Chow

2. PROVING THE GLOBAL ESTIMATES 229

In order to put (7.3) into the form of a heat equation, we need ~ \lk Rm.
Since for any tensor A, the commutator [Vk, ~] A is given by

we find that

k

[ vk, ~ J A = vk ~A - ~ vk A = L v.i Rm* vk-.i A,

.i=O

k

:t vk Rm=~ vk Rm+ L vJ Rm *vk-.i Rm.

.i=O

Substituting this formula into (7.3) and recalling that

2 (~A,A) = ~ JAJ^2 - 2 JVAJ^2 ,

we obtain

(7.4a)

k

(7.4b) + L v.i Rm*vk- j Rm*Vk Rm.

j=O

Now applying identity (7.4) in the case k = m, we get a differential inequality

:t JVm RmJ^2 :S ~ JVm RmJ^2 + f Cmj IVJ Rml -lvm-J Rml · IVm RmJ,

.i=O

where the constants Cmj depend only on j, m, and n. The inductive hy-

pothesis then gives an estimate

:t JVm RmJ^2 :S ~ JVm RmJ^2 + (cmo + Cmm) K JVm RmJ^2

(

~ Cj Cm-j ) K2 J m I

+ ki Cmj tJ/2 t(m- j)/2 \7 Rm

:S ~ JVm RmJ

2

+ K ( c:ri JVm RmJ

2

+ t~7 2 K JVm RmJ)

on the time interval 0 < t :S a/ K, where the constants c:n and c::i depend

only on m and n. Completing the square on the right-hand side and using

the fact that (a+ b)^2 :S 2 (a^2 + b^2 ) , we obtain Cm depending only on m and

n such that

(7.5) :t JVm RmJ^2 :S ~ JVm RmJ^2 + CmK (1vm RmJ^2 + ~~).

As in the case m = 1, we shall not try to control JVm RmJ^2 directly from

this equation; instead, we define