- BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 359
Integrating by parts and throwing away a negative boundary term, we have
(26.71)r H (x, T; y, v) dμg(T) (x)
}M-B9(y,A,;:r=v)< c c;/2 roo Vol9 B9 (y, p) e-D(~~v) .!!-.._ ( p2 ) dp
- }Ay:r=v Volg Bg (y, P,.) dp D (T - v).
Now by the volume comparison theorem, for p E [VT - v, oo),Vol9 B9 (y, p) < Vol_6 (p)
Vol9B9 (y, ffl -Vol_n~l ( ffl'
so thatr H (x, T; y, v) dμg(T) (x)
JM-B9(y,A,;:r=v)l
oo Vol K (p) 2
< 2C c;/^2 -n=I e-f;"" _j}_dp- Afo Vol _ K. n-1 (JI) DO" '
where O" ~ 7 - v. It is not difficult to see that, as a function of A, the
RHS of (26.71) tends to 0 as A-+ oo uniformly in O" E (0, T]. Basically, the
exponential quadratic decaying term beats the exponential linear growing
term from the volume comparison.
(2) We leave this as an exercise. DFrom combining Lemma 26.28 with Lemma 26.14, we immediately ob-
tain
COROLLARY 26.29. There exist constants A< oo and c > 0 such that
(26.72) r H(x,T;y,v)dμg(T) (x) 2: c
}B9(y,Ay:r=v)
and(26. 73) r H(x,T;y,v)dμg(v) (y) 2: c.
j B9 ( x,A,;:r=v)
We have the following lower bound along the diagonal (see Lemma 5.6
in [26]).
LEMMA 26.30 (Lower bound for the heat kernel along the diagonal).There exists a constant c > 0 such that for any y E M and 0 :S v < T :S T
(26.74)
c
H (y,T;y,v) 2: (. ~.