1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. THE t;;-NONCOLLAPSED CONDITION 85

HINT: Estimate the higher derivatives of w (r).

2. The fl;-noncollapsed condition

In this section we recall the definitions of asymptotic scalar curvature

ratio and asymptotic volume ratio, which will be used in the next chapter

to study the geometry of K-solutions. We introduce the notion of K-solution

with Harnack and we discuss a characterization of the K-noncollapsed at all

scales condition for ancient solutions by the boundedness of the entropy.

2.1. Invariants reflecting the geometry at infinity.

Perhaps the most interesting case of K-solutions is when the underlying

manifold is noncompact. In this case we are interested in the geometry at

spatial infinity of the solution.

2.1.1. Asymptotic volume and scalar curvature ratios.

Given a complete noncompact manifold (Nn, h) with Re ;:::: 0 and p E N,

the asymptotic volume ratio is defined by

(19.7) AVR(h) ~ lim VolB(p, r) E [O, 1],

r--+oo Wnrn

where Wn is the volume of the unit Euclidean n-ball.

Recall that the Bishop-Gromov volume comparison theorem says that

if (Nn, h) is a complete Riemannian manifold with Re ;:::: 0 and p E N, then

Vol~~p,r) is a nonincreasing function of r for all r > 0. The fact in (19.7)

that AVR(h) :::; 1 follows from this.

Given a complete noncompact manifold (Nn, h) and p E N, the asymp-

totic scalar curvature ratio is defined by

(19.8) ASCR(h) ~ limsup R(x) · d(x,p)^2.

d( x ,p )--+oo

It is easy to show that the limits in the definitions of AVR( h) and

ASCR ( h) are independent of the choice of p E N.

REMARK 19.15. Note that if the scalar curvature of (Nn, h) is un-

bounded, then ASCR (h) = oo. Essentially, ASCR (h) < oo if and only
if the scalar curvature has quadratic (or faster) decay from above. In par-

ticular and more specifically, if ASCR (h) < oo, then for every c; > 0 there

exists p < oo such that

R(x) · d(x,p)^2 < ASCR (h) + c;

for all x EM -B (p,p).