1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

354 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

The following says that, in an integral sense, the heat kernel decays

exponential quadratically fast in space; this result follows from Lemma 26.17

and Lemma 26.21.

LEMMA 26.23 (Heat kernel has average exponential quadratic decay).

Let (Mn, g ( r)), T E [O, T], be a complete noncompact evolving Riemannian

manifold and let H be the minimal positive fundamental solution to (26.4).

Suppose that g (0) is such that Rcg(O) 2:: -K. Then there exist constants

C = C (n, T, K, sup IRij I) < oo and D = D (T, sup IRij I) < oo such that

(26.61)

r 2 (d;(o)(x,y)) c

}AH (x,T;y,v)exp D(r-v) dμ 9 (^7 )(x):S ( ~)'

M Vol 9 (o) B 9 (o) y, - 4 -

(26.62)

{ 2 (d;(O) (x, y)) C

JM H (x,T;y,v)exp D(r-v) dμg(v) (y) :S ( ~).

Vol 9 (o) Bg(O) x, - 4 -

PROOF. As in the discussion following Theorem 24.40, let {Oi}~ 1 be an

exhaustion of M and let Hni (x, r; y, v) denote the corresponding Dirichlet

heat kernels. By Lemma 26.17, we have

c

Hni(x,r;y,v):SH(x,r;y,v):S ( )'

"'\TI vo g(O) B g(O) y, -2-~

so that

lo. H~i (x, r; y, v) dμg(T) (x)

i

:S c( ~) r Hni(x,r;y,v)dμg(T)(x)

Vol 9 (o) Bg(O) y, - 2 - lni

CeC1T

(26.63) < -------

• Vol 9 (o) Bg(O) (y, 7)

since fni Hnidμ 9 ( 7 ) :S JM H dμ 9 ( 7 ) :::; e^01 (^7 -v) :::; e^01 T by (26.37).

Now given y EM and v E [O, T), define Gi: Dix (0, T-v]-+ [O, oo) by

Gi (x, O') ==::. Hni (x, v + O'; y, v)

for i large enough. Define f : (0, oo) -+ (0, oo) by

.. f (O') =;=. CeC1T .1 Volg(O) Bg(O) ( y, 2 v0) '

so that