1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

404 8. APPLICATIONS OF THE REDUCED DISTANCE

along ')'V, and hence we can use (7.48) to get^4

Iv (7)12 g*(T) < -:--e6(n-l)ec-1/2 + 12 (n ~c - 1) 2 (e6(n-l)e - 1).

Therefore by Holder's inequality,

(t ltv {T)lg,(r) dT)

2

:::; foT' T-l/^2 dT foT' VT li'V (T)l~*(T) dT

:::; 2vfcr 1

71

T-l/^2 JV (T)J~*(T) dT

< ( 2 vfcr)2 (e6(n-l)ec-1/2 + C~c (e6(n-l)e - l)).

• 12(n-1)^2
  r2
  <-. - 16

We get the last inequality by choosing c1 :::;! such that

4 (e6(n-l)et1/2 + c~.S-

2

(e6(n-l)e _ l)) < _!__

12(n-1)^2 -15

for all t E [O, c1]. Hence

If we also require ci :::; 6 ~, then T^1 :::; cr^2 :::; ~:. Since J Rmg* ( x, T) J :::; r\

for all x E Bg(o)(P,r) and TE [O,r^2 ], we have g(x,T) 2::: ~g(x,O) for
TE [0,T'] and x E Bg(o)(P,r). Hence

dg(O) ('yv (T') ,p) :::; foT' li'v (T)lg(O) dT:::; ~ foT' li'v (T)lg(T) dT

3r r

<-<-. - 8 2

This contradicts /'V ( T^1 ) E 8Bg (O) (p, r /2). The lemma is proved. 0

We now give a proof of Proposition 8.28.

(^4) Note that (3 (£T) =i= 1'V (£T (^2) /4) satisfies ~~ = V. In (7.48) we take Co= nr-; (^1) , T = cr (^2) ,
and C2 = ~~.