1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

140 3 0. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

where

(

D = 2 exp 2nM (Woo-t)-- (r- 1) ). vw -a d2 _ 9oo _ (t o~) (q,poo_ )

. r-l 2 oo oo Woo - t

2n

2

M ( -)J_,. 2Cn ( ) J-r

+--w 3 00 -t^2 +--w 32 00 - a 00 2

2 -r - r

if r > 1 and

D

-'-

2

(2 (w 00 - to))nM .. 1 _ d~ 00 (to) (q,poo)

-:- V Woo O'.oo ------

W00 - t Woo - t

+ 4n^2 M (w 00 - l) ~ + 2Cn (woo - O'.oo) ~

if r = 1. If we assume that q is contained in a compact set JC C M 00 , then D

depends on JC and not on q.

By the mean value theorem for integrals, there exists t i E [f, w 00 -^3 I ] such that

3,

1 2 1 r^00 -^4 1 2 4D

.Jwi - ti IYi (ti)lg,(t;) :::; Woo - 3; -t Jr .Jwi - t IYi (t)lg;(tl dt < ~

since f :::; w 00 - c. Thus

(30.25)

i.e., we h ave estimated IV Lil~, at some time ti E [f,w 00 -^3 :J.

STEP 4. E stimating [\7 L i l 9 , using th e ODI. Let I b e the maximum subinterval

of [f, ti] containing f and such that [Yi (t)lg,(t) 2 1 for all t EI. Then by (30.23),

for all t E I, we have that

(30.26)

I

dt d IYi (t)lg,(t)^2 I :::; E IYi (t)lg^2 ,(t)'

where

. Cn Cn

E 7 3 , + -( ) < oo.

(~)2r-2 ~ r

We conclude from (30.26) on the interval I and from (30.25) that

(30.27) ~ IV L i I~, ( i ( q) ,f) = IYi (f) I~,(£) :::; const,

We have a similar estimate for the time derivatives of the reduced distances.

LEMMA 30 .8 (Uniform bounds for the time derivative of Li on compact sets).

Under the assumptions of Lemma 30.4, for any compact set JC C M 00 and any