1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 363

By the monotonicity formula (26.55) in the proof of Lemma 26.21, we have

(JM u[e~idμ) (~) ~ 1~ (JM u[e~idμ) (s)

= JMxiidμ

=Vol (Ui)

fort> 0, where xui denotes the characteristic function of Ui. We conclude

from this and (26.82) that

e d^2 (U1,U2) 4t^11 H (x, y, t) dμ (x) dμ (y) ~Volz^1 (U^1

1 ) Volz (U 2 ),

U1 U2

which is (26.78). D

REMARK 26.33. Davies' result may be used to give an alternative proof

of a pointwise upper bound for the heat kernel on complete Riemannian

manifolds satisfying the parabolic mean value inequality (see Theorem 1.2

in Li and Wang [120]).

PROBLEM 26.34. Is there a 'Ricci fl.ow' version of Davies' estimate?

3. Heat balls and the space-time mean value property

The heat equation may be considered as an 'averaging process'. This idea
is especially important in physics. A beautiful illustration of this averaging
process is the space-time mean value property (MVP) for solutions to the
heat equation. We consider first the classical case of Euclidean space and
then Riemannian manifolds.

3.1. Space-time mean value property on Euclidean space.
We consider the elliptic case of the Laplace equation and the parabolic
case of the heat equation.
3.1.1. Euclidean mean value property for the Laplace equation.
In Euclidean space m:n, the mean value property for solutions of the
Laplace equation is given by the following. Let Wn denote the volume of the
unit n-ball and let B(O, r) denote the ball of radius r centered at a point
0 E ffi.n.

THEOREM 26.35 (MVP for Laplace equation on m:n). If u is a 02 solution
to l::i.u = 0 in a domain 0 C IRn, then

(26.83) u(O) = ~ ( u(x)dμE (x)

Wnr JB(O,r)

for any 0 E 0 and r > 0 such that B(O, r) c 0.