300 24. HEAT KERNEL FOR EVOLVING METRICS
Since
f ( C')£ r£ (1 - a) r (1 - ,6) ( r - aY(l-a)-,6
e=o r (1 + £ (1 - a) - ,6)_ -,6^00 ( C'r (1-a) (r - (J')^1 -a)e
-r(l-,B)(r-(J') 2= r(1-,s+e(1-a))
£=0(the RHS converges since a < 1), the series I:£: 0 1Mko+£ (xo, r; z, (J')I con-
verges uniformly on 8M x 8M x IRf.
Case 2. There exists ki = k EN such that -,6k 1 2:: 0 and -/k < 1-2a
for all 1 :S k :S ki. Then (24.105) implies
,6 -"(kIMk 1 l(xo,r;z,(J')'.SCk 1 (r-(J')- kidT^1 (xo,z)
- -"(k
::; ckl dT l (Xo, z)'
where Ck 1 ~ Ck 1 r-,6k1. Substituting this in (24.98a), we have
· - 2a-1-"(k
IMk1+1 (xo,r;z,(J')I::; ck1+1dT^1 (xo,z)for some Cki +1 < oo. Iterating this, we have
- .l(2a-1)-'Yk
IMk1H(xo,r;z,(J')I::; ck1+£dT^1 (xo,z)
for all£ EN U {O}. Hence, by taking£ sufficiently large so that£ (2a - 1) -
/ki 2:: 0, this brings us back to Case 1. This completes the proof of the claim
(i.e., (24.84)) and hence Lemma 24.31.Now we make a couple of observations. Since I:£: 0 IMkH (xo, r; z, (J')I
converges uniformly on 8M x 8M x IRf, by (24.91) we have
where
f (7 d(J' f Mk (xo, r; z, (J') b (z, (J') dμ 9 (a) (z)
k=l lo laM
= (7 d(J' { M 00 (xo,r;z,(J')b(z,(J')dμ 9 (a)(z),
lo laM
00
Moo (xo, r; z, (J') ~ L Mk (xo, r; z, (J').
k=l
Hence, using (24.88), we see that 'lj; 00 (defined by (24.83)) may be written
as
00
'I/Joo (xo, r) - 2b (xo, r) = L Ak (xo, r)
k=l