1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

350 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Now fix 0 < TI < T2 < T and 0 < RI < R2 such that NR 2 (K) C int (D).

Given T E ( T 2 , T) and J > 0, let a-= T + J. By the monotonicity formula

(26.55) we have

i

v 2 ( x, TI ) exp ( - - (-PA^2 (x) ) ) dμg(T1) ( ) x

n 2~ T+J-n

l

(26.57) 2:: v^2 (x, T2) exp ( - _ (-PA^2 (x) ) ) dμg(T2) ( ) x

n 2~ T+J-~

2:: fR 2 ( T2)

since PRz (x) = 0 for x E D-NR 2 (K). On the other hand, the LHS of (26.57)
is

since dg(O) (x, K) ::::; RI for x E NR 1 (K) and since exp (- 2 d 0 (~T+8-T1^2 (x) ) ) ::::; 1

for all x En. We conclude that

::::; exp (- _ (('--Ri)' ) ) { v^2 (x, TI) dμg( 71 ) (x)

2Co T + J - TI JNR1 (JC)

(

(R2 - RI)

2

<exp ) ---^1

• 2Co ( T + J - TI) f (TI)

by (26.52). Now we take T . T2 and 6 . 0 to obtain

(26.58) 1R 2 (T2) - IR 1 (TI) ::::; -J:-exp ( (~^2 -RI)

2

).

f (TI) 2Co (T2 - TI)

Now given T and R, for i EN U {O} define

T

Ti ::i= --:-,

'Yi