1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. NO FINITE TIME LOCAL COLLAPSING 253

PROOF. We leave it as an exercise to trace through the definition of /'i,-

noncollapsed and verify that the lemma follows from the scaling properties:

B9(x, r) = Ba29(x, ar), J Rma29(y)J = a-^21 Rm9(y)J, and Vola29 Ba29(x, ar)

= anVol9 B9(x, r). D

The next lemma says the property of being /'i,-noncollapsed below the

scale p is preserved (stable) under pointed Cheeger-Gromov limits.

LEMMA 6.48 (!1;-noncollapsed preserved under limits). Let {(Mk, 9k, Ok)}

be a sequence of pointed complete Riemannian manifolds. Suppose that there

exist !1; > 0 and p > 0 so that each (Mk, 9k) is !1;-noncollapsed below the

scale p. Furthermore assume that (Mk, 9k, Ok) converges to (M~, 900 , Ooo)

in the pointed Cheeger-Gromov C^2 -topology. Then the limit (M 00 , 900 ) is

!1;-noncollapsed below the scale p.

PROOF. This is because the distance function, the curvature, and the

volume all converge under the limit. In particular, suppose x E M 00 and

r <pare such that JRm9 00 (y)J S r-^2 for ally E B9 00 (x,r). Then for every

EE (O,r), there exists k(c:) EN such that JRm9k(y)J::; (r-c:)-^2 for all

y E B ?ik ( x, r -E) and for all k 2: k ( E). Since each 9k is !1;-noncollapsed below

the scale p, we have Vol9k (B9k (x, r-c:)) 2: !1; (r - c:t for all k 2: k (c:). Taking

the limit as k ----+ oo, we have Vol9 00 (B9 00 (x, r - c:)) 2: !1; (r - c:t. Letting

E----+ 0, we then conclude that Vol9 00 (B9 00 (x,r)) 2: !1;rn as desired. D

Recall that if Re 2: 0 on a complete Riemannian manifold (Mn, 9),

then for p EM fixed, Vol~~p,r) is a nonincreasing function of r. When Mis
noncompact, the notion of !1;-noncollapsed at all scales is closely related to
another invariant of the geometry of infinity called the asymptotic volume
ratio, which we now define. ·

DEFINITION 6.49 (Asymptotic volume ratio). Let (Mn, 9) be a com-

plete noncompact Riemannian manifold with nonnegative Ricci curvature.
The asymptotic volume ratio is defined as the limit of volume ratios by

(6.80) AVR (9) ~ lim VolB (p, r) < oo,

r--->oo Wnrn

where Wn is the volume of the unit ball in JRn. We say that (M,9) has

maximum volume growth if AVR(9) > 0.

REMARK 6.50. The asymptotic volume ratio is independent of the choice
of basepoint p E M.

EXERCISE 6.51 (AVR > 0 implies noncollapsed). Show that if (Mn,9)

is a complete noncompact Riemannian manifold with Re 2: 0 and AVR (9) >

0, then fJ is !1;-noncollapsed on all scales for !1; = Wn AVR (9).