1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

396 8. APPLICATIONS OF THE REDUCED DISTANCE

PROOF. We justify the differentiation under the integral sign in equality

(8.17). Consider the difference quotient for the reduced volume integrand:

( T + h)-n/2e-£(q,r+h) dμg(r+h)(q) _ T-n/2e-£(q,r)

<I>( h)::::::: dμg(r)(q)

q,T,. h

Note that

(8.29) ddV ( T) = ( 4n)-n/^2. lim r <I>( q, T, h)dμg(r) ( q)'

T h~~.

so that the time-derivative of V ( T) exists if the limit on the RHS exists. At

any point (q, T) where f, is differentiable (e.g., for each T, a.e. on M), we

have

lim <I>(q, T, h) = T-n/^2 exp [-f (q, T)] (:_!!.__ - ~e + R).

h-+0 2T UT

Recall by Lebesgue's dominated convergence theorem that if we can show

there exists a function ]! ( q, T) such that for T > 0 there exists Er > 0 where

(8.30) J<I>(q, T, h)J .:::; \]! (q, T) on M

for h E (-En Er), and JM]! (q, T) dμg(r) (q) < oo, then

lim r <l>(q, T, h)dμg(r) (q) = r lim <l>(q, T, h)dμg(r) (q).

h-+0 JM JM h-+0

Thus, provided we have (8.30),

d

d V (T) = (4n)-n/^2 r lim <l>(q, T, h)dμg(r) (q)

T JM h-+O

= (4n)-n/^2 r T-n/^2 exp [-f (q, T)] (-!!.__ - {)f, + .R) dμ (r) (q)

JM 2T 8T g

.:::; o,

where the last inequality follows from (7.151). This is the reduced volume
monotonicity formula.
To see (8.30), we first observe that

is a locally Lipschitz function of h near h = 0, for T > 0 fixed. (Note that
1(q, T, 0) = 0.) Hence 1(q, T, h) is an absolutely continuous function of