1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BOUNDS OF THE HEAT KERNEL FOR AN· EVOLVING METRIC 351

where"( is as above with f being ("(, A)-regular, and define.

Ri ~ ( ~ + i ~ 2 ) R,

so that both Ti and Ri are decreasing in i and

To= T, Ro= R,

We have

_lim IRi h) = _lim r e-^2 CoTiu^2 (x, Ti) dμg(r) (x)

i-+oo i-+oo ln-NRi (JC)

=0

since limr\,,O u (x, T) = 0 locally uniformly for x E 0 - K (which implies
uniformly for x E 0 - Ni R (K)). Therefore
2
00
IR (T) = L (IRi (Ti) - IRH 1 h+i))
i=O

by (26.58).
Since J is ( 'Y, A)-regular, we have

By this, Ri -Ri+l = (i+ 2 )(i+ 3 ) R 2: (i:S)2, and Ti - Ti+I = ~i+i T, we obtain

1

00

( • ( j (To))^1 i+1 R

2

)

h(T)~ f(T)~exp (i+l)log Af(T1) -2Co(i+3)4('Y-l)T.

Let

.. 'Yi+l
B= mf 4.

. iENU{O}(i+3) (i+2)('Y-1)

Since 'Y > 1, we have B > 0. Then

(26.59) IR (T) ~ e ~~ f exp ((i + 1) (log (A~ (To)) - B_R

2

)) •

f (T) i=O f (T1) 2CoT

We estimate the quantity on the RHS of (26.59) as follows.