1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

370 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

PROOF. (1) The integral is finite. Let (xo, to) EM x (0, T]. By (26.103)

and (26.105), we have

E;(xo, to) c { y EM: d^2 (xo,y) < DTlog (~~~)} x {to - T}

for TE (0, to] and r E (0, oo). In particular, for each such T and r, the heat

ball slice E;(xo, to) is bounded.

Define f E (0, oo) by

C1rn = f(5to).

Then log(~~;;))= 0, so that E~^0 (xo,to) is empty. Moreover, by the con-

tinuity off, for any r E (0, f) there exists Tr < to such that E;(xo, to) is

empty for T > Tr.

Since IV' log HI 2 ( xo, y, T) is bounded for T away from 0, the integral in

(26.106) is finite over any subdomain of E;(xo, to) where Tis bounded away

from 0.

To obtain the finiteness of the integral in (26.106) when T is close to

zero, we note the following gradient estimate due to Hamilton [91], which

is related in spirit to the Li-Yau differential Harnack estimate [121]

(26.107) IV log HI^2 :S -;=-log C ( Tn/B 2 H )

(see Corollary E.34 in Part II). Now by the asymptotics of H(x 0 , y, T) as

T--+ 0, we have that the integral on the right-hand side of (26.106) is finite

for T close to zero. (Exercise: Prove this.)^7

(2) MVP via a conservation law. Let Er ~ Er(xo, to) and let Er,t 1 -

Er n {t =ti}. Given r > 0, as in (26.90) we let

(26.108) 'l/Jr(Y, T) = logH(xo, y, T) + nlogr.

Again, the heat ball may be expressed as

Er= {(y, t) : 'l/Jr(Y, T (t)) > O}.

It is easy to check that 'I/Jr (y, T ( t)) satisfies

( 26.109 ) {it+ 8'1/Jr .6..'l/Jr = -IV''l/Jrl^2 ·

Given any smooth function u(x, t), define

(26.110) I (r) ~ r~ J J u(y, t)IV''l/Jrl^2 (y, T(t)) dμdt.
Er
Having discussed a sufficient condition for the integral I to be well defined,
i.e., when H satisfies (26.104), we may proceed to compute :fr!.
Without loss of generality, we may change the time interval to [-T, O]
and assume that to= 0, so that T(t) = -t. Let

(26.111) J (r) ~ rn I (r).

(^7) See Theorem 10 and Remark 11 in §3 of [57].