1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. HEAT KERNEL FOR AN EVOLVING METRIC 341

where CE (0, oo) is as in (25.84) and where we.used fo:::; 1 and (26.31). By

applying (26.33) and (26.18) to (26.32) and integrating by parts, we obtain

d~ JM <p (x, T) HD (x, T; y, v) dμg(r) (x)

(26.34) ~JM ( b.x,r'P (x, T) - 2VCA) HD (x, T; y, v) dμg(r) (x)

• C1 JM <p (x, T) HD (x, T; y, v) dμg(r) (x),

where C1 is as in (26.18). Since there exists C' E (0, oo) depending only on

n, r, and ma:X:Mx[O,T] IRcl such that

b.x,r'P (x, T) ~ -C',

by applying (26.27) to (26.34), we obtain

d~ JM <p (x, T) HD (x, T; y, v) dμg(r) (x)

~ - ( C' + 2VCA) eCi(r-v) - C1 JM <p (x, T) HD (x, T; y, v) dμg(r) (x).

Now (26.28) follows from integrating this in time and from the fact that

lim r <p (x, T) HD (x, T; y, v) dμg(r) (x) = 1.

r'\,v} M

EXERCISE 26.11. Derive another lower bound for

JM HD (x,T;y,v)dμg(r) (x)

D

using the method in the derivation of (22.41) in the proof of Lemma 22.9.

Finally, we note that the semigroup property holds for the Dirichlet heat
kernel.

LEMMA 26.12. The semigroup property holds for the Dirichlet heat ker-

nel.

PROOF. Using Exercise 26.2, the lemma follows from the same technique
as when the compact manifold has no boundary. D

1.4. Elementary properties of the heat kernel on a noncompact

manifold.

As before, let g ( T) be an evolving complete Riemannian metric on a
noncompact manifold Mn satisfying f 7 9ij = 2Rij for T E [O, T] and let

H (x, T; y, v), where x, y E M and 0 :::; v < T :::; T, be the heat kernel for