314 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICSN ow smce · Lii=l '\''00 (n+2)-i --;;;:- _ - n 2 , we h ave
(25.36)^6 = e^0 s(i+^2 v'K'ro)~ (Vol_g B9 (xo, ro))-~
oo 2 l(n+2ri
X il ( (i+~+14i+3) (26Q4i+2ro2)7i:)2 n.Since cn+i4 0 4 > - (^1) ' we have
( 1 + c0+14i+^3 ) ( 2C04i+^2 r()^2 ) ~ :::::: 2 n!
2
00+^3 ( 4i+^3 ) n!2
r~~'where(25.37)
(^00) (. n+2) Hn!^2 ri "= (' )(n+2)-i+l
Cn ~ il ( 4i+3) n = 2L-i=l i+3 1t < 00
is a constant depending only on n.
Summarizing, we have shown that
(25.38)
where
A eCs(1+2VKro)% -(n+3)n A
C = 1 C 0 4 Cn < oo,
ro (Vol_g B9 (xo, ro)) 2
Co= - Coe 2AT is as in (25.5), and Cs and Cn A ~ 2-4-cn n+2 depend only on n.
Slightly generalizing the above argument, we have the following.
LEMMA 25.6. For any r E [ro, 2ro) and/ E (1, 2] such that 1r :::::; 2ro, we
have
(25.39)where
A ecs(1+1VKr)% -(n+3)n A n
C(r)~ 1 C 0 4 Cn(r-1)-4<oo,
r (Vol_g B_g (x 0 , r)) 2(25.40)
where Cn depends only on n.EXERCISE 25.7. Prove Lemma 25.6.HINT: Instead of (25.29), define for i EN
ri ~ r ( 1 + ~:})
so that ri = 'yr and limi-+oo r i = r.