1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

294 24. HEAT KERNEL FOR EVOLVING METRICS

Claim. The series in (24.83) converges uniformly on 8M x [O, T] to a

continuous function and we may exchange the sum and integration:

(24.84)

1

T 1 8ii OO

= d(J -

8

• (xo, Ti z, ()) L Ak (z, ()) dμg(u) (z).
  O 8M Vz,u k=l

The claim, (24.83), and summing (24.82) from k = 1 to oo imply

(24.85) ~

1

7

'I/Joo (xo, T) = 2b (xo, T) + 2 d(J^1 -£)-8H (xo, Ti z, ())'I/Joo (z, ()) dμg(u) (z),

0 8M UVz,u

i.e., 'ljJ 00 is a solution of (24.78). By (24.76) we conclude

(24.86)

1

7

U'¢= (x, T) = - d(J^1 -£)-8H (x, Ti z, ())'I/Joo (z, ()) dμg(u) (z)

0 8M UVz,u

is the desired solution to Lemma 24.31.

Now we prove the claim. First we rewrite the Ak. Fork= 1 we have

1

7

Ai (xo, T) = 2 d(J^1 -£)-8H (xo, Ti z, ())Ao (z, ()) dμg(u) (z)

0 8M UVz,u

= 2 r d(J { Mi (xo, Ti z, ()) b (z, ()) dμg(u) (z)'

lo laM

where

is defined by

(24.87)

8ii

Mi (xo, Ti z, ()) ~ 2-

8

• (xo, Ti z, ()).
  Vz,u
  By induction we can show that

(24.88) Ak (xo, T) = 2 r dO" { Mk (xo, Ti z, ()) b (z, ()) dμg(u) (z)'

lo laM

where the functions

Mk : 8.M. x 8M x JR~ -+ JR

are defined recursively by
(24.89)

1

Mk+l (xo, Ti z, ()) ~ 2 T1 -8ii

8

--(xo, Ti w, p) Mk (w, Pi z, ()) dμg(p) (w) dp

u 8M Vw,p