1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

372 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

deduce

:/ ~ ~ L (!,,,,, ( \O,~u + u (

8

ti' + 1vM)) d1}t

= -n 1°^1 ( 1/Jr-au + U-a1/Jr) dμdt

r _ 00 Er,t at at

+ "2.1° 1 uJ\71/JrJ^2 dμ dt

r -oo Er,t

+ "2.1° r 1/Jr (/}..u - ~u) dμ dt

r -oo }Er,t t

(26.113) = -J n + -n 1° 1 1/Jr ( /}..u - -au) dμdt.

r r -oo Er,t at

Note that, since 1/Jr = 0 on aEr,t, we have

so that

Therefore, from (26.113) we have, for almost every r, that

(26.114) _E_ dr (.!___) rn = _!!__ rn+ 1 jj 1/Jr (f}..u - au) at dμ dt.

Er

In the special case where u(x, t) is a solution to the heat equation, we have

d

-I=:O.

dr

Note that for any 5 > 0, the sequence of (incomplete) Riemannian

manifolds {(B(xo,5),x 0 ,i^2 g)}iEN converges as i--+ oo, in the C^00 pointed

Cheeger-Gromov sense, to (ffi.n, 0, gcan), where gcan denotes the standard

Euclidean metric. Moreover, from the scaling property

and the heat kernel asymptotics we can show that

lim I(r, xo, g) = u(xo, to).

r->0