1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 355

Let Vol _ _K_ n-1 B (r) denote the volume of a ball of radius r in the simply-

connected n-dimensional space form with constant sectional curvature -n1!._ l ,

where K ;:::: 0 (so that Re :::::::: -K). We claim that f is ('I', A )-regular with

'Y = 4 and

Vol _ _K_ B (Vf)

A= n-1

Vol _ _K_ B ('If).

n-1

Indeed, since Rcg(O) ;:::: -K, we have by the Bishop-Gromov volume com-

parison theorem that

f (ai) _ Volg(O) Bg(O) (y, ~)

f (aif 4) Volg(O) Bg(O) (y, v;1)

Vol _ K B (Vf1)

< n-1

• Vol _ K B (~)
  n-1
  Hence, by Lemma 26.21 with n = ni, u = Gi, and K = {y }, we have

(26.64)

{ 2 d~(o) (x,y) 4AeCu

Jni Gi (x, a) e D<7 dμg(v+u) (x) :S f (a / 4 )'

Taking the limit as i -t oo (one may first integrate over compact regions),

we have

{ d~(o)(x,y) 4AeC(r-v)

}MH

2

(x,T;y,v)e D(r-v) dμg(r)(x) :S f((T-v)/ 4 )

4Ce(C+C1)T_-n=-r-_~~ Vol K B( -1')

< _____ Vol_ ___,_n=-r_----,~ K B( "7)

Volg(O) Bg(O) (y, 7)

This proves (26.61). Similarly, one may prove (26.62); we leave this as an
exercise for the reader. D

EXERCISE 26.24. Justify that Lemma 26.21 can be applied to prove

(26.64) even though the fundamental solution Gi (x, a) is a 8-function dis-

tribution centered at y at time a = 0.

2.1.3. Pointwise upper bound of the heat kernel for an evolving metric.

One may combine the estimates in Lemma 26.23 with the semigroup
property to prove the following upper estimate for the heat kernel, which
extends an estimate of Li and Yau to the case of a time-dependent metric
(see Theorem 5.1 in [26]; compare with Corollary 3.1 of Li and Yau [121]).

THEOREM 26.25 (Upper bound of the heat kernel for an evolving metric).

There exists a constant C3 < oo depending only on n, T, K, and sup IRijl