358 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICSwhere C4 depends only on n, K, and T. Thus (26.67) impliesC3Cf exp ( - 4 ;:cr2)H(x,r;y,v)::; \T, 0 l-B-( g g x, (J ).
The corollary follows from taking D ~ C5 ~ 2C4. DEXERCISE 26.27. Verify that (26.68) is true.2.2. Lower bound of the heat kernel for an evolving metric.
We now derive a lower bound for the heat kernel with respect to an
evolving metric.
We begin with the following, which is Lemma 5.5 in [26].
LEMMA 26.28 (L^1 -norm of H concentrates at y as r -v-+ 0). We have
the following.
(1)(26.69)(2)
(26.70)r H(x,r;y,v)dμg(T) (x)-+ 0
j M-B9 (y,Ay'T=V)as A -+ oo uniformly in 0 ::; v < r ::; T.
{ H(x,r;y,v)dμ 9 (v) (y)-+ 0
}M-B9(x,Av'T=V)
as A -+ oo uniformly in 0 ::; v < r ::; T.PROOF. (1) UsingH(x,r;y,v)::; ( (;=;)