1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. HEAT KERNEL ON NONCOMPACT MANIFOLDS 293

Note that

(1) compared to formula (24.76) in the interior, we have the extra

'jump' term ~1,b (xo, T),

(2) the integrand 8 ~ii (xo, T; z, O") has a singularity at (z, O") = (x 0 , T),

(3) for any Xo E oM,~nd E > 0

lim ( {)H (x,T;z,O")i,b(z,O")dμg(cr)(z)dO"

x-txo J(aMx [O,r-i::])U((EJM-B(x 0 ,i::)) x [r-c:,r)) OVz,cr

1

oH

= -

0

-(xo,T;z,O")i,b(z,O")dμg(cr) (z)dO",

(8Mx [0,r-i::])u((EJM-B(xo,i::)) x [r-i::,r)) Vz,cr

SO that the jump term is due solely to the singularity at (xo, T).

EXERCISE 24.34. Prove Lemma 24.33. HINT: See the notes and com-
mentary at the end of this chapter.

By Lemma 24.33, in order to prove Lemma 24.31, it suffices to find a
continuous function 1,b : oM x [O, T] ---+IR. satisfying the equation
(24.78)

1,b (xo, T) = 2b (xo, T) + 2 r dO" r :;jj (xo, T; z, O") 1,b (z, O") dμg(cr) (z)

Jo laM uVz,cr

for all Xo E oM and T E [O, T], where b : oM x (0, T] ---+ IR. is continuous
and lifilr\,O b (x, T) = 0 for all x E oM.
To prove existence, we iterate this equation; namely, consider the se-
quence { 'iPk}~_ 1 of functions on oM x [O, T] defined recursively by 'iP-1 = 0
and
(24.79)

1

'iPk (xo, T) ~ 2b (xo, T) + 2 T dO" in -'1-aii (xo, T; z, O") 'iPk-1 (z, O") dμg(cr) (z)

0 EJM UVz,cr

for k E N U { 0}. Let

(24.80) Ak (xo, T) ~ 'iPk (xo, T) - 'iPk-1 (xo, T).

Since 'iP-1 = 0, we have

(24.81) Ao (xo,T) = 1,bo (xo,T) = 2b(xo,T)

and by (24.79) we have the recurrence relation

1

7

(24.82) Ak(xo, T) = 2 dO" i -'1-{)H (xo, T; z, O") Ak-1 (z, O") dμg(cr) (z)

O EJM UVz,cr

for k E N. Define 1,b 00 : oM x [O, T] ---+ IR. by

(24.83)

00

'iPoo · (xo, T) ~ ~ L..,, Ak (xo, T) = k-too lim 'iPk (xo, T).

k=O