124 20. COMPACTNESS OF THE SPACE OF KrSOLUTIONS
In §3, we prove that for any n 2: 3 the collection of n-dimensional K,-
solutions with Harnack is compact modulo scaling. As a special case, using
a result of §Ll in Chapter 19, we obtain Perelman's fundamental result that
the collection of 3-dimensional 11;-solutions is compact modulo scaling. Com-
pactness is one of the cornerstones of 3-dimensional singularity analysis. It
enables one to probe the geometry of 11;-solutions by taking limits of pointed
sequences. The proof of Perelman's compactness theorem exploits the di-
chotomy between the properties of no local collapsing and AVR = 0.
In §4 we discuss a rather easy but important consequence: scaled de-
rivative of curvature estimates for 3-dimensional 11;-solutions. We also state
some optimistic conjectures concerning the classification of ancient solutions
in low dimensions.
1. ASCR and AVR of /\;-solutions
In this section we consider the asymptotic scalar curvature ratio ASCR
and the asymptotic volume ratio AVR of 11;-solutions (see subsection 2.1 of
Chapter 19 for the definitions of ASCR and AVR). To oversimplify, the
reason we consider these invariants is that curvature, volume, and distance
are important in the study of the Ricci flow. The AVR is an 'infinite radius'
version of the volume ratio seen in the definition of 11;-noncollapsing. The
ASCR is related to the notion of scaled curvature bounds.
1.1. 11;-solutions with Harnack have infinite ASCR and zero
AVR.
Let (Mn,g(t)) be a complete noncompact 11;-solution with Harnack and
fix a point p EM (see Definition 19.27). Recall that the AVR and ASCR
are defined by (19.7) and (19.8), respectively. Denote
AVR(t) ~ AVR(g (t)) and ASCR(t) ~ ASCR(g (t)).
The following result was proved by Perelman in §11.4 of [152].
THEOREM 20.1 (11;-solution with Harnack has ASCR = oo,AVR = 0).
For any complete noncompact 11;-solution with Harnack with n 2: 21 we have
ASCR(t) = oo and AVR(t) = 0
for all t E ( -oo, OJ.
As a consequence of the proof of Theorem 20.1 we have the following.
COROLLARY 20.2 (Any ancient solution with bounded Rm 2: 0 must
have AVR = 0). Let (Mn, g ( t)), t :::; 0, be a complete noncom pact non fiat
ancient solution to the Ricci flow. Suppose g(t) has nonnegative curvature
operator and
sup I Rm I (x,t) < oo.
(x,t)EMx(-oo,OJ
Then the asymptotic volume ratio is zero, i.e., AVR(t) = 0 for all t.
