1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

248 23. HEAT KERNEL FOR STATIC METRICS

obtain

(23.105)

1

fPPN

.. (x, z, t - s) G (z, y, s) dμ (z)
M 8x18x1

1

8PN 8G

= - --j (x,z,t-s)

8

i (z,y,s)dμ(z)

M 8xz Xz

- f

8

P~ (x,z,t-s)G(z,y,s) (

8

8

i Jdetgk.e) dx;···dx~,

JM 8xz Xz

where [ 8 ~ 1 Jdetgk.e[ :SC in B (x,inj (g)). As in the proof of Lemma 23.26,

by applying (23.88) to (23.105), we have

(23.106) {

82

.PN. (x, z, t - s) G (z, y, s) dμ (z) :SC (t - s)-°',

JM 8x18x1

where a E (!, 1). Thus the integral on the RHS of (23.92), i.e.,

1

t1 82pN 1t 82JN

.. (x,z,t-s)G(z,y,s)dμ(z)ds =
8

i

8

. (x,y,s,t)ds

o M 8x~8x~ o x xJ

is the sum of two integrals each of which converges absolutely.

Now by Lemma 23.26, i.e., (23.90), proving Lemma 23.27 is equivalent

to showing that

(23.107) -^8 1t 8JN 1t 82JN

8

. -
8
. (x, y, s, t) ds =
8

.

8

. (x, y, s, t) ds
xi o xJ o xi xJ
exists. Let the path Ii be as above. By the mean value theorem, we have
(23.108)
1; ~ (ri ( h) , y, s, t) ds - J; ~ (ri ( 0) , y, s, t) ds -1t 82 JN ( t) d
h o 8 xi. 8 xJ. x, y, s, s

1

t ( 82JN 82JN )

=

8

.

8

. (ri (h*), y, s, t) -
8

.

8

. (x, y, s, t) ds
0 xi xJ xi xJ
for some h* contained in the interval from 0 to h.

Since 1; a~~~~j (x, y, s, t) ds is a sum of integrals which converge abso-

lutely, for any c: > 0, there exists r5 ( c:) > 0 such that

l

t 1::;N. (x',y,s,t),ds<c:

. t-8(c) X xJ

independent of x' E M. On the other hand, given r5 (c:) > 0, there exists

'TJ > 0 such that if Jhl < 'T], then

I

8

2

JN 82 JN I c

8xi8xJ (Ii (h*) 'y, s, t) - 8xi8xJ (x, y, s, t) < t