1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

392 8. APPLICATIONS OF THE REDUCED DISTANCE

(ii) For any VE TpM and 0 < 7 < T,

(8.23) ( 47r 7 )-n/2 e-£bv(r),r) £ Jv ( 7 ) :S 7r-n/2e -IVl~co,p).

Hence, even for a complete solution on a noncompact manifold, the

reduced volume is well defined.

PROOF. (i) Recall from (7.139) that

(..5.._log£Jv) d7 (7) :S ~ 27 - ~7-~K. 2

From this and (8.21), we compute

d~ [ ( 47r7 )-n/2 e-£('-yv(r),r) £ Jv ( 7) J

=(47r7)-n/^2 e-£bv(r),r)_cJv(7) (-~ - df, + ..5.._log.CJv)

27 d7 d7

:S 0.

(ii) It follows from (8.22) that for any 0 < 7 < 7v, we have

( 47r 7 )-n/2 e-£bv(r),r) £ Jv ( 7 )

(8.24)

:S lim (47rT1)-n/^2 e-£bv(ri),ri)_cJy(71)

71 --+0+

= lim [(47r71)-n/2 £Jv (71)] e-limT1->0+£C1'v(r1),r1)

r1--+0+

• 7r-n/2e-IVl2


• '

where in the last equality we have used (7.138) and (7.97). If 7 2: 7V, then
the statement is obvious. D

An immediate consequence of the above lemma is the following funda-
mental result: the monotonicity of the reduced volume.

COROLLARY 8.17 (Reduced volume monotonicity). Suppose (Mn,g(7)),
7 E [O, T], is a complete solution to the backward Ricci flow with the curva-
ture bound IRm (x, T)I :S Co< oo for (x, T) EM x [O, T]. Then

(i) limr--+O+ V (T) = 1.

(ii) The reduced volume is nonincreasing:

(8.25) v ( 71) 2: v ( 72)

for any 0 < 71<72 < T, and V(7) :S 1 for any 7 E (O,T).

(iii) Equality in (8.25) holds if and only if (M,g (7)) is isometric to

Euclidean space (~n, gJE) , regarded as the Gaussian soliton.

PROOF. (i) From equation (7.131) it follows that

lim n(p o) (7) = TpMn.

r--+O '