1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

386 8. APPLICATIONS OF THE REDUCED DISTANCE

then

1-.'!.!!

0 < V - ( g, "'ar 2) S K,^2 K,

(47rt^12 + -. wn

Hence if a< 2/n, then assumption (8.12), as"'---> 0, implies V (9, "'ar^2 ) --->

0.

1.3. The noncompact case and the asymptotic volume ratio.

Consider the case where ( JVt.n, g) is complete and noncompact with Reg 2: 0.

It is natural to believe that the limit as T ---> oo of the static reduced volume

V (9, r) is related to the limit as r ---> oo of the volume ratios; as we now

show, this is indeed the case. Inequality (8.10) implies

lim V(9,r) s inf VolB(p,r) =AVR(9),

r->oo r>O Wnrn

where

AVR(9) : lim VolB(p,r) = lim A(s)

r->oo Wnrn S->00 nWnSn-1

as in (6.80). Next we show the opposite inequality. Since, by (8.8),

A (r) 2: nwn AVR (9) rn-l,

we have for all T > 0,

V (9, r) = 1

00

(47rr)-n/^2 e-s

2

/^4 r A (s) ds

2: nwn AVR (9) 1

00

(47rr)-n/^2 e-^82 /^47 sn-lds = AVR (9).

Therefore the limit, as T tends to infinity, of the static reduced volume is

the asymptotic volume ratio.

LEMMA 8.10 (Asymptotic limit of Vis AVR). If (JVt.n,9) is a complete

noncompact Riemannian manifold with Reg 2: 0, then

lim V (g, r) = AVR (9).

T->00

REMARK 8.11. When n = 2, (8.9) says for any r > 0

(8.13) V- (A ) Area B (p, r) ( r

(^2) - r2)
g, T S 2 -
4

• e 4r •
  1fr T

Note that the function F (x) ~ x+e-x, x 2: 0, is an increasing function and

its minimum value of 1 is attained at x = 0. In particular, as in (8.10) the

upper bound in (8.13) improves as T increases (for fixed r > 0).

2. Reduced volume for Ricci fl.ow

In this section (Mn, g ( T)) , T E [O, T] , will be a complete solution to the
backward Ricci fl.ow satisfying the curvature bound I Rm ( x, T) I S Co < oo
for (x,r) EM x [O,T].