136 3. THE COMPACTNESS THEOREM FOR RICCI FLOW
Note that, using (3.7), we can rewrite V - Vk = r - rk as a sum of terms
of the form V 9k· When i = 1, we can bound
[vN-l (V - Vk) Rck[
by a sum of terms of the form I vN-j 9k I I vj Re k I ' 0 s j s N - 1. When
2 S i S N, we can bound
[vN-i (V - vk) vi-^1 Rck[
by a sum of terms of the form [vN-i-Hlgk[ [VjVJ;^1 Re k[, Os j s N - i.
We can also bound
[vjvi-^1 Re k [ = J ((V - Vk) + Vk)j vi-^1 Re k J
by a sum of terms which are products of Jv£+i-l RckJ, 0 S f S j, and
several [Vcgk[, 1::; f::; j. By the assumption of Lemma 3.11, the induction
assumption and the equivalence of l·I and l·lk, we get
[VNRck[ s C'Jv JvN 9k[ + C'Jj-.
Now we turn to bounding [ V N 9k J. Since g does not depend on t,
f) N N
fJt V 9k = -2V Re k
and
f) N 2 I f) N N ) I f) N 1
2
fJt JV 9k[ = 2 \ fJt V 9k, V 9k S fJt V 9k + I V N 9k[^2
= 4 [VNRck[
2
+ JvN 9k[
2
s ( 1+8 (C'Jv)
2
) JvN 9k[
2
+ 8 (C'/j-)
2
.
Integrating the above differential inequality of [VN 9k[^2 , we get (compare
with (7.4 7))
[VN 91c[2 (t) S e(i+s(c~)2)(t-to) (JvN 9k[2 (to)+ 8 (C'Jj-)2 ).
1+8 (C'Jv)
2
This implies
[vN 9k (t) [ s CN,o,
and the induction proof is complete, as well as (3.4) for the q = 0 case.
Nate that the above proof of bounding JV N Re k [ can be used to show that
J VPV% Re k J , [ VPV%Rk [ , and [ VPV% Rm k J are bounded independent of k.
eq eq-1
When q;:: 1, then atq VPgk (t) =VP atq-l (-2Rck (t)). Using the evo-
lution equation of the curvature Rm k ( t), we know that I gt: VP 9k ( t) I is
bounded by a sum of terms which are products of
J ~vP^1 V%^1 Rm k J ( t) , J VPV% Re k [ , and J VPV%Rk J.
8q -
Hence we get I etq VP 9k ( t) I S Cp,q. 0