1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 275

for a E (0, 1). Taking a E a, 1), we have

I~~~ (x, r; y, v; O")I ~JM I c;::~ (x, r; z, O") G (z, O"; y, v)/ dμg(u) (z)

~ c (7 - 0")-0I

and the improper integral

1

T 1 {)pN 1T {)JN

v M oxi (x,r;z,O")G(z,O";y,v)dμ 9 (u)(z)dO"= v {)xi (x,r;y,v;O")dO"

on the RHS of (24.32) converges absolutely. Finally, similarly to (23.90), we
may show that

{) f,T {)JN

~ uxi (PN * G) (x, r;y,v) = v ~ uxi (x, r;y, v; a") dO"

exists. D

Let B 9 (T) (x,r) = {y EM: d 7 (y,x) < r}. Analogous to Lemma 23.27,
we have

LEMMA 24.17 (Second space derivatives of a convolution with PN). Un-

der the same hypotheses as Lemma 24.16, for (x, r) EU x (0, T] such that

B 9 ( 7 ) (x, ~ inj(g (r))) CU,

we have PN * G is C^2 with respect to the space variable x and the second

space derivatives of PN * G are given by
(24.34)

[)2 (PN * G) 17"1 [)2 PN

oxi{)xj (x,r;y,v) = v M {)xi{)xj(x,r;z,O")G(z,O";y,v)dμ 9 (u)(z)dO",

where {xi} are geodesic coordinates centered at x with respect to g ( T).

SKETCH OF PROOF. Let JN be as in (24.33). We have

[)2JN

0

.

0

xi xJ. (x,r;y,v;O")

= { ( {)~ PN. -

02

.PN.) (x, r; z, O") G (z, O"i y, v) dμg(u) (z)

IMA {)xi x ox^1 x oxi z ox^1 z

1

[)2pN

+ .. (x,r;z,O")G(z,O";y,v)dμ 9 (u)(z).

M ox~ox{

(1) Similarly to (23.102), we have

a2pN.. - a2pN.. ( X, T,. z, O" )<c(1 _ + d;.(x,z))c T - O" )-~ e -~f;~';j.

oxi x ox^1 x oxi z ox^1 z T - (J"

(2) Similarly to (23.106), we have for a E (~, 1)