1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

388 8. APPLICATIONS OF THE REDUCED DISTANCE

is roughly (since the sphere has small curvature for N large) the volume of

the hypersurface M x SN x fr} in M and its volume can be computed as

Vol9 8B9 (p, v121Vf)

~ r dμ9M(Tw) (x) A T;;:12dμsN (y)

1as 9 (p,~)

~Vol( SN, 9sN) JM ( Vr - 2 ~L(x, Tw) + O(N-^2 )) N dμ 9 M(r)

~ WN ( v121Vf)N JM (1-

2

;VrL(x,f) + O(N-^2 )) N dμ 9 M(r)'

where WN is the volume of the unit sphere SN (recall that 9sN has constant

sectional curvature 1/ (2N), i.e., radius ../'iN). Observing that

lim (1 -

1

JTL(x, f) + O(N-^2 )) N

N-+oo 2N T

(

= lim 1----L(x 1 1 f) )N = e --2V¥^1 -L(xr) ' = e-£( x,T -)

N-+oo N 2JT ' '

one can prove

(8.15)

Vol9 ( 8B9 (p, Y2iif))

( Y2iif)N+n

= (2N)-n/2WN (JM 7-n/2e-R(x,r)dμgM(7) +O(N-1)).

In particular, we obtain the geometric invariant

JM f-n/2e-R(x,r)dμgM(7)

for f E (O,T).

EXERCISE 8.13. Make the above arguments rigorous (especially the ap-

proximations) and in particular prove (8.15).

2.2. Definition of Perelman's reduced volume. Thus we are led

to the following.

DEFINITION 8.14 (Reduced volume for Ricci fl.ow). Let (Mn, g (T)), TE

[O, T] , be a complete solution to the backward Ricci fl.ow with bounded

curvature. The reduced volume functional is defined by

(8.16) v (T) ~JM (41fTrn/^2 exp [-.e (q, T)] dμg(T) (q)

for T E ( 0, T).