1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. REDUCED VOLUME FOR RICCI FLOW

analysis (see Corollary 15 on p. 110 of [313]), we have

ip(q,T,h) = ~ (cp1(q,T,h)-cp1(q,T,0))

= ~ rh ~ [(T + h)-nf2e-£(q,T+h) dμg(T+h) (q)] dh

h lo ah dμg(T) (q)

(8.31) =h}o^1 rh ( -2(T+h)._ n OT(q,T+h)+R(q,T+h) [)£ )

X (T + h)-~ e-.f.(q,T+h) dμg(T+h) (q) dh.

. dμg(T) (q)
By Lemma 7.59, we have

(

2 ,-, T d 9

2

(0) (p, q) nCof)

(8.32) exp[-£ (q, f)]:::; exp -e-'-'^0 4f + -

3

• ,

397

which decays exponentially quadratically in terms of the distance function.

Also we have e. I..

07 dμ 9 (^7 ) T='f (q) = R (q, f) dμg('f) (q), so that we have the

following bounds for the volume form:

-nColhl < dμg(^7 +h) (q) < nColhl

e - dμg(T) ( ) q - e.

By Lemma 7.59, we also have

/) ( -) 200 r d;('f) (p, q) nCof

.c,qT <e +--

' - 4f 3 '

so it follows from Lemma 7.60(ii) that

I

• [)£ I C1
  0

(q,T) _:::;--=--(£(q,f)+Af)

T T=T T

(8.33) -< C1 - (e2C0Td;('f) 4-(p, q) + (nCo 3 + A) f) '

. T T

where A < oo. Hence, for f = T + h, we have

I

--n - -[)£ ( q T -) + R ( q T -) 1--!! T 2e -R.(q ' 7) dμg('f) (q)

2f OT ' ' dμg(T) (q)

(8.34) -< -+-2f n C1 f ( e^200 rd;('f) 4f (p, · q) + (nCo -+ 3 A) T -) +no c,

x T __ !! 2 exp ( -e -20^0 Td;(O) (p, q) nCof) nCol'f-TI

47

+ -

3

• e ·

On the other hand, by the curvature lower bound and the Bishop-Gromov
volume comparison theorem,

(8.35) Vol B 9 ( 7 ) (p, dg(T) (p, q)) :::; C4eC^4 dg(r)(p,q)