1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. HEAT KERNEL FOR AN EVOLVING METRIC 339

In particular,

(26.25)

(a:: + ,6,,y,vHD) (x, Ti y, v) + (R - Q) (y, v) HD (x, Ti y, v) = 0,

(26.26) lim HD (x, Ti ·, v) =Ox.

V/'T

PROOF. The proof of Lemma 26.3 holds for the Dirichlet heat kernel
since the integration by parts in (26.13) is still valid. D

Similarly to Lemma 26.5, we have

LEMMA 26.10 (L^1 -norm of Dirichlet heat kernel is bounded). The Dirich-

let heat kernel HD for Lx, 7 , on a compact manifold (Mn, g ( T)) with nonempty

boundary, satisfies:

(1)

(26.27)

for any y E int (M) and 0 :S v < T :S T, where C1 is as in (26.18).

(2)

(26.28) JM HD (x, Ti y, v) dμ 9 ( 7 ) (x) 2:: 1-C (T - v)

for any y E int (M) and TE [O, T], where C < oo depends only on

A~ sup JRij (x, T)J 9 ( 7 ) < oo,

Mx[O,T]

(26.29)

(26.30) r ~ -

4

1

min { min d 9 ( 7 ) (y, 8M), min inj g( 7 ) (y)},

TE[O,T] TE[O,Tj

C in (25.84), C1 in (26.18), and maxMx[O,T] JRcJ.

PROOF. (1) We compute

! JM HD (x, T; y, v) dμ 9 ( 7 ) (x)

=JM (a:: (x, Ti y, v) + R (x, T) HD (x, T; y, v)) dμg(T) (x)

=JM (,6,,x,THD) (x,T;y,v)dμg(T) (x)

+JM (R-Q) (x, T) HD (x, T; y, v) dμg(T) (x)

:Sr Vx(HD)(X,Tiy,v)dμ 9 ( 7 )(x)

JaM

• C1 JM HD (x, Ti y, v) dμ 9 ( 7 ) (x),