1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 365

EXERCISE 26.36 (Mean value inequality when Re 2: 0). Show that if

(Mn, g) is a complete Riemannian manifold with Re 2: 0 and if f : M ---+

( -oo, O] is a nonpositive C^2 function with !:lf 2: 0, then for any 0 E M and

0 < r < inj(O)

(26.89) t (o):::; ~ r fdμ.

Wnr JB(O,r)

HINT: By the Laplacian comparison theorem, we have fl (p^2 ) < 2n,

where p (x) ~ d (x, 0). Defining

7/Jo(r) ~ ~ r f(x)dμ (x)'

Wnr JB(O,r)

we find that gr'l/Jo(r) 2: 0 (note that p(x)^2 - r^2 :S 0 for x E B(O,r)). (See

for example Proposition 1.142 in [45] for another proof.)

REMARK 26.37. By taking f = -1 in (26.89), we have

VolB (0, r) :S Wnrn,

which, of course, is a special case of the Bishop-Gromov volume comparison
theorem.

3.1.2. Space-time Euclidean mean value property for the heat equation.

Analogous to the mean value property for harmonic functions there is
a space-time mean value property for solutions to the heat equation on
Euclidean space, which we now present.
First we discuss the heat kernel and the associated heat balls. Recall
that the Euclidean heat kernel is defined by

n _ lx-yl^2

HE(x, t; y, s) ~ (47r (t - s))-2 e 4(t-s)

for x,y E ]Rn and -oo < s < t < oo. Given r E (O,oo), we define the heat

ball of radius r based at (x, t) as the following superlevel set of the heat

kernel

Er(x, t) ~ { (y, s) E IRn x (-oo, t) : HE (x, t; y, s) > r~},

which is an open convex subset of IR.n x (-oo, t). Note that the point (x, t)
is actually at the zenith (or top) of the heat ball Er ( x, t); the nadir (or

bottom) of the heat ball is the point (x, t - ~;). The boundaries of the heat
balls are the level sets of the fundamental solution of the heat equation.
Let
(26.90) 'I/Jr (x, t; y, s) ~ log(rn HE)

I 1

n x-y^2

= n log r -

2

log ( 47r ( t - s)) -

4

( t _

8

) •