1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. HEAT KERNEL ON NONCOMPACT MANIFOLDS 295

for k E N and by (24.87). Indeed, assuming (24.88) holds for some k EN,

we have

Ak+l (xo, r) = 2 (7 {

8

8

jj (xo, r; z, O") Ak (z, O") dμ 9 (u) (z) dO"

lo leM Vz,u

1

= 2 T in -aii

8

• (xo, r; z, O") dμ 9 (u) (z)
  0 8M Vz,u

x 2 {u { Mk(z, O"i w, p) b ( w, p) dμ 9 (p) ( w) dpdO".
lo leM
Switching the order of integration for p and O" and switching names for w
and z, we have

Ak+l(xo,r)=4 (7 (7 { {

8

8

H (xo,r;w,O")Mk(w,O";z,p)b(z,p)

lo lP leM laM vw,u

X dμg(u) (w) dμg(p) (z) dO"dp

=4 r1

7

{ {

3

BH (xo,r;w,p)Mk(w,p;z,O")b(z,O")

lo u leM leM Vw,p.

x dμg(p) (w) dμg(u) (z) dpdO"

=2 (7 dO" { Mk+i(xo,r;z,O")b(z,O")dμ 9 (u)(z).

lo leM

This completes the proof of (24.88).

By (24.88), the claimed formula (24.84) is equivalent to the facts that
the series

(24.90) f Ak (xo, r) = 2 f (7 dO" { Mk (xo, r; z, O") b (z, O") dμ 9 (u) (z)

k=l k=l lo leM

converges and

(24.91)

L OO 1T dO"^1 a;;-aii (xo, r; z, O")

k=l o 8M z,u

x fu dp { Mk(z,O";w,p)b(w,p)dμ 9 (p)(w)dμ 9 (u)(z)

lo leM

1

7

= dO"^1 - 8H

8

• (x 0 , r; z, O")
  O BM Vz,u

xf r dp { Mk(z,O";w,p)b(w,p)dμ 9 (p)(w)dμ 9 (u)(z) ..

k=l lo leM

To see both of these facts, we estimate !Mk I· First note that

(24.92)