- BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 353
by (26.52). Second, we compute
r 2 d~(O) (x,/C)
}/Ci v (x, r) e Dr dμg(T) (x):::; C4
~~2
r V^2 (x, 7) dμg(T) (x)
... ln-int(N 2 i-l R(JC))since JCi c D - int (N 2 i-1 R (JC)) and d 9 (o) ( x, JC) ~ 2i R for x E JCi. Now
1
v^2 (x, r) dμ 2A ( 4i-1R2)
S1-int(N^9 (^7 ) (x) :::; - exp -B -
2 i-lR(JC)) f (r(y) 2CoT
by (26.60).
Combining all of the above estimates, we obtain
r 2 d~(O) (x,/C)
Jn v (x, r) e Dr dμg(T) (x)
. 1 R^2 2A^00 4iR^2 ( 4i-l R^2 )
< --eDr + Lel5T exp -B -
- f(r) f (r/y) i=l ·. 2Cor
1 R^2 2A ~ R^2 4i(1-BP)
= -_-eDr + - ~ eDr BCo •f (r) f (^7 /'y) i=l
R2
Taking R so that eDr = 2, we conclude
1
d~(O) (x,/C) 2 2A^00 4i (i-Bp)
v^2 (x, r) e Dr dμg(T) (x) :S; ---+ _ L 2 sco.
n f ( r) f ( r I I) i=lNow choose D =^1 6f 0 , so that
r 2. d~(O) (x,IC)
Jn v (x, r) e Dr dμg(T) (x)
2 2A f 2 _ 4 i
:::; f (r) + f (r/1) i=l
8+A< -
- 4f (r/1)
4A
<---- - f(r/1)
since 2::~ 1 2_^4 i:::; ~' J (r/1):::; J (r), and A~ 1. Thus we have
{ 2 d~(o) (x,IC) 4AeCT
Jou (x,r)e Dr dμg(T) (x) :S; f (r/i)'where C ~ 2Co ( 1 - 1-^1 ). This completes the proof of Lemma 26.21. 0