1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. REDUCED VOLUME OF A STATIC METRIC 383

where we used JV'dJ = 1 a.e. and the Laplacian comparison theorem, i.e.,

d.b..d::::; n - 1. Nc>te that the Laplacian comparison theorem is equivalent to

the Bishop-Gromov volume comparison theorem. D

It is useful to keep the following simple examples in mind when pondering

Lipschitz continuous sub-and super-solutions of the heat equation.

EXAMPLE 8.6 (Heat equation on 51 ). Let M^1 = 51 = ffi./ (2nZ) with

the standard metric g = dB^2. Consider the function

e2

f (B, t) = t + 2' e E (-n, n] and t ER

For each fixed e, f ( e, ·) is a smooth (linear) function of time and, for each

fixed t, we have f (·, t) is Lipschitz on 51 and C^00 except ate= n. Moreover

f is a solution to the heat equation almost everywhere. In particular,

of a^2 f

at ( e, t) = ae2 ( e, t) = i

for all e E 51 - { n} and t E R On the other hand,

d

d r f (e, t) de= 27r > o.

t Js1

This is consistent with the fact that f is not a subsolution of the heat

equation in the weak sense (as we shall now see, f is a supersolution). Note

that
of
ae (e, t) = e

for all e E ( -n, 7r) and t E ffi. (and ~~ is undefined for e = 7r). In particular,

for each t E ffi., ~~ (·, t) has a jump discontinuity at e = n. In the sense of

distributions, we have ~~ (e, t) = e and

IJ2f

(8.5)

8

e 2 (·, t) = 1 - 2n. c57f,

where 67f is the Dirac 6-function centered at e = n. Hence, in the sense of

distributions,

of a^2 f 02 f

at = ae 2 +^2 n ·^6 1f 2:: 3e2 ·

EXERCISE 8.7. Prove (8.5).

SOLUTION TO EXERCISE 8.7. For any C^2 function <p: 51 --+ ffi. we have

f e

2

<p"(B)dB= e

2

tp'(e)\7f - { B<p'(B)de

Js1 2 2 -7[ Js1

= f <p(B)dB-B<p(B)J:_1f

Js1

= f <p(B)dB-2n<p(n).

Js1