1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

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