1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

6. GRADIENT SHRINKERS WITH NONNEGATIVE RICCI CURVATURE 31

for any continuous piecewise c= function ( : [O, r (x)] -t JR satisfying ( (0)

( (r (x)) = 0. Now define (to be

( (s) = { ~

r(x)-s

ro

if 0 ::::; s ::::; 1,

if 1 < s ::::; r ( x) - ro,

if r ( x) - ro < s ::::; r ( x) ,

where ro is to be chosen below. We have

r(x)

Jo (^2 Rc(r^1 , 1^1 )ds::::;(n-l)(ri]^1 +1).

From this we obtain

(27.142)

1 r(x) Re (r', 11 ) ds::::; (n - 1) (r 01 +1) + fo

1

(1 - (^2 ) Re (r', 11 ) ds

i

r(x)

+ (l-(^2 )Rc(r',1^1 )ds

r(x)-ro

2

::::;(n-l)(r 01 +1)+- max_ Rc(V,V)

3 VETyM, IVl=l, yEB 6 (1)

i

r(x)

+ r(x)-ro R (r(s)) ds

since Re;::: 0.

We now estimate J;(~x/-ro R ( 1( s)) ds. Since V R = 2 Re (V f) and Re 2: 0, we

have for all y E M

IV RI (y) ::::; 2 IRc l (y) IV fl (y) ::::; 2R (y) ( ~r (y) + Vf(O)) ,

using (27.30). Thus, if y EM satisfies r (y) ;::: 1, then

(27.143) IVlnRl(y)::::; (1+2Vf(O))r(y).

Now choose r 0 =^4 S(~)l) ::::; 1. Suppose that s E [r (x) - ro, r (x)]. Using

r (1(s)) ;::: 1, the assumption that R (x) ::::; 1, and (27.143), we compute that

lnR(r(s))::::; ln Rl~~~))

r(x)

::::; Js IVlnRI (r(s))ds

::::; (1+2Vf(O)) ir(x) r (r(s)) ds

r(x)-ro

::::; (1+2Vf(O))ror (x)

::::; 4 (n - 1) (1+2V](O)).

That is, for s E [r (x) - ro, r (x)],

R (r(s)) ::::; e4(n-1)(i+2v7(0)).