1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

344 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Integrating this, we find that for any TI < T2

eG1(^72 -v) JM ¢(x)H(x,T2;y,v)dμ 9 ( 72 ) (x)

- eCi(Ti-v) JM¢ (x) H (x, Ti; y, v) dμ 9 ( 71 ) (x)

= 1~ d~ (eCi(T-v) JM ¢(x)H(x,T;y,v)dμg(T) (x)) dT

> -Cn 1T2 e2C1(T-v)dT.

• R Tl
  Taking the limit as R --+ oo, we obtain

JM H (x, T2; y, v) dμ 9 b) (x) 2: e-Gi(T^2 -^71 ) JM H (x, Ti; y, v) dμ 9 ( 71 ) (x).

Since lim 71 '\,v JM H (x, Ti; y, v) dμ 9 ( 71 ) (x) = 1, we conclude

JM H(x,T2;y,v)dμ 9 ( 72 ) (x) 2: e-Gi(T^2 -v).

D

The semigroup property for closed manifolds may be extended to the

complete case.

LEMMA 26.16 (Semigroup property-noncompact case). Let (Mn, g(T)) 1

T E [O, TL be a complete noncompact evolving manifold. The minimal posi-

tive fundamental solution H to Lx, 7 u = 0 satisfies

(26.41) H (x, T; y, v) =JM H (x, Ti z, p) H (z,p; y, v) dμg(p) (z)

for any x, y E M and 0 :S v < p < T :S T.

PROOF. As in the discussion following Theorem 24.40, let {Di}~ 1 be an

exhaustion of M by smooth domains with compact closure such that Di C

ni+l and let H [Ii ( x' T; y' v) denote the Dirichlet heat kernel of ( ni' g bi).

By Lemma 26.12, the semigroup property holds for the Dirichlet heat kernels

Hoi, that is, for any x, y E ni and 0:::; v < p < T:::; T we have

Hoi (x,T;y,v) = r Hoi (x,T;z,p)Hoi (z,p;y,v)dμg(p) (z)

Joi

:S JM H (x, T; z, p) H (z, p; y, v) dμg(p) (z)

for all i since H oi :::; H. Taking the limit as i --+ oo yields

H (x, T; y, v) :::; JM H (x, T; z, p) H (z, p; y, v) dμg(p) (z).