- BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 345
On the other hand, fix a compact domain/(, c M. We have
l H(x,r;z,p)H(z,p;y,v)dμ 9 (p) (z)
= _lim { Hni (x, r; z, p) Hni (z, p; y, v) dμg(p) (z)
i-+oo }JC
::::; _lim Hni (x, r; y, v)
i-+oo
= H(x,r;y,v).
Since this is true for any K, we concludeJM E (x, r; z, p) H (z, p; y, v) dμg(p) (z)::::; H (x, r; y, v).
metric 2. Upper and lower bounds of the heat kernel for an evolving
metricD
The mean value inequality, the differential Harnack estimate, the bound-
edness of L^1 -norm, and the semigroup property may be used to derive
bounds on the heat kernel.
2.1. Upper bounds of the heat kernel for an evolving metric.
We first obtain an upper bound for the heat kernel H (x, r; y, v) in terms
of volumes of balls which is not sensitive to the distance between x and
y. Then we prove that H decays exponential quadratically in the space
variables in a weighted L^2 sense. Finally we demonstrate that H decays
pointwise exponential quadratically.
2.1.1. A rough upper bound for the heat kernel.
The following result is obtained by combining the mean value inequality
(at a suitable spatial scale) with the above lemma on J H dμ ::::; C.
LEMMA 26.17 (Rough upper bound for H). Let (Mn,g(r)), TE [O,T],
g, Co, Ci, C2, and C3 all be as in Theorem 25.2. Let H (x, r; y, v) be the
minimal positive fundamental solution to (26.4). Then for all x, y EM and
o:::;;v<r:::;;T
(26.42) H(x,r;y,v):::;;min{ ( v=)' ( v=)}'
Vol_g B_g x, - 2 - Vol_g B_g y, - 2 -where C < oo depends only on n, T, Co, C1, C2, C3. In particular, we have
the following upper bound along the diagonal:
c
(26.43) H(x,r;x,v)::=;; ( vr-v)·
Vol_g B_g x, - 2 -