1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

292 24. HEAT KERNEL FOR EVOLVING METRICS

THEOREM 24.32 (Existence of Dirichlet heat kernel). The function H

defined by (24. 75) is the fundamental solution for Lx, 7 satisfying the Dirich-

let boundary condition:

H (x, r; y, v) = 0 on 8M x int (M) x (0, oo) ,

lim H ( ·, r; y, v) = 5y for y E int (M).

T'\,V

In the remainder of this subsection we give a proof of Lemma 24.31.

Without loss of generality we may assume that v = 0. Given a continuous

function

'ljJ : 8M x [O, T] -+ ~'

define

U1f; : int (M) x [O, T] -+ ~

by

(24.76)

1

7

U1f; (x, r) ~ - d(J"^1 -£:)-aH (x, r; z, o-) 'ljJ (z, o-) dμg(O") (z),

0 BM UVz,O"

where dμg(O") denotes the induced volume (n - 1)-form of 8M and where

Vz,O" denotes the outward unit normal to 8M at z, all with respect tog (o-).

Given x E int (M), we have

lim sup IV'g(O") iii (x, r; z, o-) ~ 0.

T-tO (z,O")EBM x [0,T)

Thus we have

T-tO lim U1f; (x, r) = U1f; (x, 0) =^0 for x E int (M)

and also (abusing notation with 'BBH Vz,r (x,r;z,r)')

Lx,TU'lj; (x, r) = - r do- r Lx,T (:ii ) (x, r; z, o-) 'ljJ (z, o-) dμg(O") (z)

Jo }BM uVz,O"

1

8H

• ~ (x, r; z, r) 'ljJ (z, r) dμ 9 ( 7 ) (z)
  BM UVz,T
  =0,

because Lx,,, ( B~~u) = B~,u ( Lx,,,ii) = 0 (since Lx, 7 and Bv~,u act on

distinct factors, they commute) and B~~T (x, r; z, r) = 0 for x E int (M)

and z E 8M.

To understand the boundary values of U1f;, we have the following.

LEMMA 24.33 (Jump relation). For any xo E 8M we have

(24.77)

lim U1f; (x, r) =^1 - 1T^1 aii

2

1/J (xo, r)- do- -£:)-(xo, r; z, o-) 'ljJ (z, o-) dμg(O") (z).

x-txo o BM uVzp

Here the limit is taken with x inside any finite closed cone C c Int(M)u{ x 0 }.