1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

266 24. HEAT KERNEL FOR EVOLVING METRICS

where Q: Jvl x [O, T]-+ ffi. is a 000 function and where A 7 = A 9 ( 7 ) denotes

the Laplacian with respect tog (T).

As a special case, we have

(24.3) Rij = Rij and Q = R,

in which case g ( T) satisfies the backward Ricci flow and

L=D * =--u^0 A 7 +R

OT

is the adjoint heat operator considered by Perelman. Note that under (24.1)
we have

(24.4)

8

OTdμ =Rdμ,

1. Heat kernel for a time-dependent metric

In this section we state the main result on the existence of the heat
kernel for a time-dependent metric on a closed manifold. We then begin
its proof by considering estimates for the transplanted heat kernel using the
metric at time T.

1.1. Statement of the existence of the heat kernel for a time-

dependent metric.

As in (23.30), let ffi.~ = {(T,v) E ffi.^2 : 0 < T-V ~ T}.

DEFINITION 24.1 (Heat kernel with respect to an evolving metric). Let
g (T), T E [O, T], be a smooth 1-parameter family of complete metrics on a
manifold Mn. We say that

H:MxMxffi.~-+ffi.

is the fundamental solution for g 7 - Ax, 7 + Q if

(1) His continuous, 02 in the first two space variables, and 01 in the

last two time variables,

(2) letting H (x, T; y, v) ~ H (x, y, T, v),

(24.5)

(:T -Ax,T+Q)H(·, ·;y,v) =0, (! +Ay,v-Q+R)H(x,T; ·, ·) =0,

(3)

(24.6) lim H ( ·, T; y, v) = 5y,

T\,,V

limH(x,T; ·,v) = 5x.

V/"T

The minimal positive fundamental solution is called the heat kernel for

:T -Ax,T+Q.

The main result proved in this section and the next section is the fol-
lowing.