1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 369

Its time slices are given by

E;^1 (xo, to) =i= Er(xo, to) n { T (t) = T1} c M x {to - T1}

for TI > 0. Equivalently, define

Er,t 1 (xo, to) =i= Er(xo, to) n {t =ti} c M x {ti},

so that

for ti <to.

In order to ensure that the integral (26.106) below, which appears in the

statement of the mean value property, is well defined, we shall make some

additional assumptions on the heat kernel H(x, y, T) which guarantee that

the time slices E; are compact. In particular, we assume that along the

diagonal we have a bound of the form

(26.104)

c

H(y,y,T):'S f(T)' (y,T)EMx(O,oo),

for some constant C < oo and some positive, continuous, strictly increasing

function f : (0, oo) --+ (0, oo) which is (!', A)-regular in the sense of Definition

26.20 with T = oo.

Under assumption (26.104) on the heat kernel along the diagonal, by
a general result of Davies and Grigor'yan (see Theorem 1.1 of [76]), there
exist constants 5, D, C 1 E (0, oo) such that the heat kernel has the upper
bound

(26.105) C1 ( d

2

(x,y))

H(x, y, T) ::::; f(5T) exp - DT

for all x,y EM and TE (O,oo).^6
Using this estimate, we shall prove the following, which is a generaliza-
tion of (26.98).

THEOREM 26.41 (Space-time mean value property for the heat equation

• manifold version). Let (Mn,g) be a complete Riemannian manifold with
  heat kernel H(x, y, T) satisfying assumption (26.104). If u: M x [O, T] --+JR
  is a classical solution to the heat equation ~~ - b.u = 0, then for any fixed

to E (0, T] there exists r > 0 such that for any r E (0, r) and xo E M, the

heat ball Er(xo, to) lies inside a compact subset of M x (0, T] and

(26.106) u(xo,to) = r~ ff u(y,t)l\7logHl^2 (xo,y,T(t))dμ(y) dt,

Er(xo,to)

where T(t) =to - t.

(^6) In fact, for any D > 4 there exists J > 0 and C1 < oo such that (26.105) holds.