1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

276 24. HEAT KERNEL FOR EVOLVING METRICS

Now (1) and (2) imply that the integral on the RHS of (24.34) is the sum
of two integrals, each of which converges absolutely. Finally, we may finish
the proof by showing that (similarly to (23.107))

-^8 1r 8JN 1r 82JN
8
. -
8
. (x, r; y, v; er) der =
8

.

8

. (x, r; y, v; er) der.
xi v xJ v xi xJ
D

Analogous to Lemma 23.29, we have

LEMMA 24.18 (Time derivative of a convolution with PN). If G E

c^0 (M x M x IRf), then PN * G is C^1 with respect to the time variable

rand

(24.35)

8

8

r (PN*G)(x,r;y,v) =G(x,r;y,v)

1

r { 8PN

+ v JM 8 r (x,r;z,er)G(z,er;y,v)dμg(a-)(z)der.

SKETCH OF PROOF. We compute the time difference quotient

(24.36)

(PN G) (x,r + h;y,v)-(PN G) (x,r;y,v)

h

= ~ (1r+h JN (x, r + h; y, v; er) der -1r JN (x, r; y, v; er) der)

11r+h

= h r JN(x,r+h;y,v;er)der

j

er JN (x, r + h; y, v; er) der - JN (x, r; y, v; er) d

+ v h er

= -h

1

1r+h JN (x, r + h; y, v; er) der + 1r

8

8

JN/ (x, r; y, v; er) der,

r v r r=r*

where r* E ( r, r + h). To establish (24.35) from taking the limit as h \,i 0
of (24.36), one shows that

hm. 1r -8JNI 1r 8JN

8

(x, r; y, v; er) der = -

8

(x, r; y, v; er) der

h'\,O v r r=r* v r

and

1

r 8JN.

v 8 r (x,r;y,v;er)der

1

r { 8PN

= v JM 8 r (x,r;z,er)G(z,er;y,v)dμg(a)(z)der.