1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

302 24. HEAT KERNEL FOR EVOLVING METRICS

THEOREM 24.40 (Existence and uniqueness of heat kernel on noncom-

pact manifolds - time-dependent metrics). Let Mn be a noncom pact man-

ifold and let g ( r), r E [O, T], be a smooth family of Riemannian metrics

on M. If Q is uniformly bounded, then there exists a unique C^00 mini-

mal positive fundamental solution HM (x, r; y, v) for the heat-type operator

Lx,T = gT - Llx,T + Q ·

The idea of the proof is to take an exhaustion {Di}iEN of M by smooth

domains with compact closure such that Di c Di+l· Let Hni (x, r; y, v)

denote the Dirichlet heat kernel of (Di, g [ 0 i), which exists by Theorem

24.32. By the maximum principle we have

(24.106)

Therefore

(24.107) HM (x,r;y,v) ~ _lim Hni (x,r;y,v) E (O,oo]

i-+oo

exists for any (x,r;y,v) EM x M x ~(note that Hni is defined at any
(x, r; y, v) for i sufficiently large). Note that the positivity of HM follows
from (24.106).
We shall show that HM is finite, C^00 , and a fundamental solution. One
way to see that HM is finite is to apply the methods used in the proof of
Lemma 22.9.

LEMMA 24.41. Given y E !1i, we have

for r E (0, T].

PROOF. We compute

d

d r Hni (x, r; y, v) dμg(T) (x)

T Joi

1

= 8H0,. ~ ( x,r;y,v ) dμ

Oi UT^9 (^7 ) (x)

= r (..6.x,T - Q) Hni (x, r; y, v) dμg(T) (x)

Jni

= r ~~fl~ (x,r;y,v)dμg(T) (x)- r QH0,i (x,r;y,v)dμg(T) (x)

Jni T,i Joi

_:S- { QHni(x,i;y,v)dμ 9 ( 7 )(x)

Joi

:S- inf 2 Q { Hni(x,r;y,v)dμ 9 ( 7 )(x),

MxMxlRT Joi