- HEAT KERNELS UNDER CHEEGER-GROMOV LIMITS 191
Integrating the above inequality on the interval [ti,~' we obtain (note that
t-ti < ~Q-^1 )
104 -1 a - 1
dg(ti)(x*,xo)::::; dg(f)(x*,xo) + S (n -1) ynQ2 · 2 Q-
1 - 1 52. - 1
::::; d 9 cfJ(x,x 0 ) +
10AQ-2 + S (n-1) ynaQ-2
(22.27)9 - 1
< d 9 cf)(x,x 0 ) +
10AQ-2,where we used (22.24) in the second inequality and we used A 2:: 1 and
a < 13 (n~l)y'ri in the last inequality. However, inequality (22.27) contradicts
(22.23). This completes the proof of Lemma 22.8 and hence Claim 2. D2. Heat kernels under Cheeger-Gromov limits
In this section we discuss the convergence of heat kernels for a sequence
of solutions to the Ricci fl.ow which converges in the C^00 Cheeger-Gromov
sense.
2.1. Pointed C^00 convergence of solutions.
A motivating setup for our discussion in the next subsections is the
following (in view of Lemma 21.15 on Cheeger-Gromov convergence).
Let (M~, 900 (t), x 00 ), t E [-T, OJ, be a pointed complete solution to
the Ricci fl.ow. Suppose that {UihEN is an exhaustion of M 00 by open
sets with x 00 E Ui for all i and suppose that 9i ( t), t E [-T, O], are (in
general, incomplete) solutions to the Ricci fl.ow on Ui such that the sequence
{ (Ui, 9i (t)) }iEN converges in C^00 to (M~, 900 (t)) uniformly on compact sets
in M 00 x [-T, OJ. Note that here the convergence is pointwise, i.e., we do
not pull back by diffeomorphisms.
Let gi (t), t E [-T, OJ, be complete solutions to the Ricci fl.ow on Mi ::J Ui
such that
9i (t) = 9i (t) on ui,
i.e., gi (t) are complete extensions of 9i (t). For each i, let
ft: Mi x [-T, 0)-+ IR+
be the minimal positive fundamental solution to
(22.28)
aft .... ....
at = -ll·th(t)Hi + Rffi(t)Hiwith limt/'O ft(., t) = 5x 00 •
The above discussion relates to Cheeger-Gromov convergence, as in
Lemma 21.15, as follows. We have the following two setups.
(1) The limit (M~,9 00 (t) ,x 00 ), the exhaustion {(Ui,9i (t))}iEN' and
the solutions { ( Mf, gi ( t)) LEN are as above (where Mi ::J Ui c
Moo)·