3. NOTES AND COMMENTARY 331

in x E M - Bg(r)(P, }g) for each T E [O, T]. Since R 2:: Jg and for each

T E [O, T] we have¢= 1 in Bg(r) (p, R), the estimate (25.100) actually holds

on all of M x [O, T].

Similarly to (25.99), we have

p (x, r) "'-2 ~ 3o u + 63 + '-;,3o ( c, + c2"i!io'))

for x E Bg(r) (p, R) and T E (0, T]. Therefore

P(x T) > __ n_ (~+Cs+ Cg +C10)

' - 2 - 3c: T R R^2 '

where the dependences of Cs, Cg, and C10 are as in the statement of Theorem

25.9. This completes the proof of Theorem 25.9. D

We remark that there is the standard issue of the distance function f =

dg(r) ( · ,p) being only Lipschitz in Bg(r) (p, 2R) and C^00 a.e. in Bg(r) (p, 2R).

There are two ways to address this issue: use a barrier function for f or use

Calabi's trick.
(1) Barrier function. Let (xo, To) E B(p, 2 R) x (0, f] be a point at which
TcpP attains a negative minimum. Then, in a neighborhood U of this point,
P < 0. Now the function J defined using lengths of paths as in subsection
1 of Chapter 17 is an upper barrier for f in a neighborhood V c U of x 0 ,
i.e., J 2:: fin V and J (xo) = f (xo). Since 'lj; is decreasing, this implies that
¢ ~ 'lj; (l) is a lower barrier for¢= 'lj; (£) in V. Since P < 0 in V, we
have
T<jJP ::S T¢P in V.
Since T<jJP attains a minimum at (xo,To), we conclude that T¢P attains a
local minimum at (xo, To). We then obtain the corresponding estimate for

P(xo, To) with the bounds for f and its derivatives replaced by the bounds

for J and its derivatives. However these bounds are by definition the same.

(2) Calabi's trick. We leave this as an exercise; see also subsection 1.13
of Appendix A in Part I.

3. Notes and commentary

For excellent modern references on geometric analysis and the heat equa-
tion, see Schoen and Yau [168] and the forthcoming book by Peter Li [118].

§1. The parabolic mean value inequality is intimately related to the
volume doubling property (or weak volume growth condition), Sobolev in-
equality (in the form of Proposition 25.4), and the upper bound for the heat
kernel; see Li and Wang [120], especially Corollary 2.3 and the discussion
in the introduction therein.
Note that the method of Moser iteration is standard and there are other
works using this technique, especially in the fixed metric case. See the