1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. HEAT KERNEL ON NONCOMPACT MANIFOLDS 291

isometric to the product of a metric on 8M with an interval; we may do

this in a way which depends smoothly on T. Third, we double the extension

of the differentiable manifold and we double the extended metric to obtain

(M,g(T)).

Now extend Q to a C^00 function Q: M x [O, T]-+ R Let

H:MxMxJRf-+JR

be the heat kernel for Lx,r ~ ffr -b.x,g(r) +Q. We shall obtain the Dirichlet

heat kernel for Lx r on ( M, g ( T)) by adding to HI a solution to

' MxMx~

the Dirichlet problem for the heat equation.

We shall prove the following.

LEMMA 24.31 (Dirichlet problem for the heat equation). Given any v E

[O, T) and any continuous function b: 8M x (v, T]-+ JR with

lim b ( x, T) = 0 for all x E 8 M,

r'\,.v

there exists a unique c^0 (C^00 in the interior) solution u: M x [v, T] -+JR

to Lx,rU = 0 with the boundary conditions

u (x, v) = 0 for x E int (M),

u (x, T) = b (x, T) for x E 8M and TE (v, T].

By the lemma, given y E int (M) and v E [O, T), we may let

fy,v: M X [v,T]-+ JR

be the solution to Lx,rfy,v = 0 with the boundary conditions

fy,v (x, v) = 0 for x E int (M),

fy,v (x,T) = -H (x,T;y,v) for x E 8M and TE (v,T].

We then define

H : M x M x lRf -+ JR,

by

(24.75) H (x, T; y, v) ~ H (x, T; y, v) + fy,v (x, T).

Given y E int (M) and v E [O, T),

lim max \.H\ (x,T;y,v) = 0

r'\,.v xE8M

and by the maximum principle we have for TE (v, T]

max\fy,v\ (x,T) SC max \.HI (x,T;y,v)

xEM xE8M

where C < oo is a constant depending only on Q and T. Therefore we have

the following.