1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 347

since T - v ST. Thus

c

H(x,T;y,v)s "{T 1 B ( VT=v )'

vo 9 9 x,- 2 -

where C depends only on n, T, K, Co, C1, C2, and 03.

We leave it as an exercise for the reader to prove the similar inequality

c

H(x,T;y,v)s ( VT=v).

Vol9B9 y, - 2 -

D

EXERCISE 26.19. Under the setup of Lemma 26.17, let n c M be a

smooth compact domain and let Ho (x, T; y, v) denote the Dirichlet heat

kernel for L.

(1) Show that for x, y En and 0 s v <Ts Tsuch that B9 (x, 7) c

n we have

(26.45)

c

Ho (x, T; y, v) S ( VT=v),

Vol9 B9 x, - 2 -

where C < oo has the same dependence as in Lemma 26.17 (and is

independent of n).

(2) Similarly, show that if B9 (y, 7) c n, then

·C

(26.46) Ho (x,T;y,v) S ( ~).

Vol9 B9 y, - 2 -

2.1.2. Average exponential quadratic decay for the heat kernel.
We shall improve our pointwise bound so that we obtain exponential
quadratic decay in the space variables.
In preparation for the next result, we give the following.

DEFINITION 26.20. We say that a continuous, strictly increasing function
f : (0, T] -+ (0, oo) is ('y, A)-regular, where/> 1 and A 2 1, if

f ( T1) < A f ( T2)

f (71/y) - f (T2h)

for all 0 < T1 S T2 < T.

The following result says that if the L^2 -norm of a solution has a rea-
sonable time-dependent bound, then an exponential quadratically weighted
L^2 -norm of the solution has a corresponding time-dependent bound. This
integral bound is a precursor to pointwise bounds. Let int (0) denote the

interior of n.