1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

352 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Case 1. log (A j~~~D -fa~; :::::; -log 2. Then (26.59) implies

----BR^2 00

I (T) < e -zcor '""'2-i-l

R - f (T) L.J i=O

1 _ B_R2

= -_-e zc 0 r.

f (T)

Case 2. log(Aj~~~D-fc~; > -log2. Then by (26.56), (26.52), and

our assumption, we have

IR (T) :::::; in v^2 (x, T) dμ 9 ( 7 ) (x)

1

<---

- f (T)

2A BR2

< _ e -zc 0 r.

f (T/"f)

In either case we have

(26.60) { v^2 (x,T)dμ 9 ( 7 )(x)=JR(T):::; _

2

A exp(-B_R

2

)

ln-NR(K) f (T/'Y) 2CoT

usmg. f(T)^1 < f(Th)^1 < f(Th) 2A m. th e fi rs t case.

STEP 3. Exponentially weighted L^2 estimate. Given R > 0, let

Ko~ {x ED: d 9 (o) (x,K):::::; R},

Ki~ {x ED: 2i-lR:::::; d 9 (o) (x,K):::::; 2iR}

for i EN U {O}. We have for any D > 0 to be chosen sufficiently large later

r 2 d;(O) (x,K)

Jnv (x,T)e Dr dμ 9 ( 7 )(x)

1

2 d;(O) (x,!C)

= v (x, T) e Dr dμg(T) (x)

Ko

00 r 2 d;(O) (x,JC).

+ L }K v (x, T) e Dr dμg(T) (x).

i=l Ki

First note that

1

2 d~(O) (x,JC)

v (x, T) e Dr dμg(T) (x)

Ko

:::::; eDr R21 v^2 (x, T) dμg(T) (x)

Ko

1 R^2

< ---eDr

• f (T)