1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. REDUCED VOLUME FOR RICCI FLOW 395

Then for any 71 < T2,

PROOF. By the definition of the .C-Jacobian we know that for any L^1

function f on M

{ f(y)dμ 9 (r)(Y) = { f(.Crexp(V))CJv(T)dx(V).

J Lr exp(D(A,r)) J D(A,r)

(We have used this change of variables formula for A = TpM in previous

sections.) We have

VA ( T2) = r ( 411"T2)-nf2e-C(1v(r2),r2) .c Jv ( T2) dx (V)

Jn(A,r 2 )

_::::; { ( 411"Tl)-n/2e-C(1v(r1),r1) .C Jv (Ti) dx (V)

Jn(A,r2)

_::::; { ( 411"T1)-n/2e-C(1'v(r1),r1) L Jv (Tl) dx (V)

Jn(A,r1)

=VA (T1).

D

The above can be thought of as a relative volume comparison theorem
for the Ricci fl.ow. This is along the lines of the generalization by Shunhui
Zhu in [384] (see also Theorem 1.135 in [111] for example).

2.4. Monotonicity of reduced volume revisited. Now we give an-

other proof of the monotonicity of the reduced volume without using the
.C-Jacobian. Recall that under the evolution equations

0

OT 9ij = 2Rij'

-f o - b.f + IV f^12 - R + -n = 0,

OT 2T

with T > 0, we have

d~ JM T-n/2e-f dμg = 0.

In comparison, by (7.146), the reduced distance f is a subsolution to the
above equation for f. We use this fact to give another proof of the mono-
tonicity of reduced volume.

THEOREM 8.20 (Monotonicity of the reduced volume: second proof). Let

(Mn, g (r)), T E [O, T], be a complete solution to the backward Ricci flow

satisfying the curvature bound !Rm (x, T)I ::::; Co< oo for (x, T) EM x [O, T].

Then for any T E (0, T), the reduced volume V (T)-is differentiable and

non increasing:

(8.28)