CHAPTER 20
Chapter 20. Compactness of the Space of /\;-Solutions
It never would come to me working on a mystery.- From "Runnin' down a Dream" by Tom Petty
Compactness theorems are fundamental in geometric analysis. In the
case of sequences of solutions of the Ricci fl.ow with curvatures uniformly
bounded and injectivity radii uniformly bounded from below, Hamilton's
Cheeger-Gromov compactness theorem yields a subsequence which con-
verges in C^00 on compact sets (see Chapters 3 and 4 of Part I). By Perelman's
no local collapsing theorem, such sequences occur in the study of finite time
singular solutions on closed manifolds when we rescale about a sequence
of points and times whose curvatures are comparable to their space-time
maximums up to those times.
On the other hand, we are interested in the geometry of finite time sin-
gular solutions in their high curvature regions, not just where the curvatures
are comparable to their space-time maximums. For example, we are inter-
ested in rescaling a singular solution about arbitrary sequences of basepoints
and times whose curvatures tend to infinity. For such a sequence, the rate
of blow up of the curvatures at the basepoints may be slower than the rate
of blow up of the corresponding spatial maximums of the curvatures. That
is, for such a sequence, the curvatures are not comparable to their space-
time maximums. Moreover, the study of high curvature regions starts with
singularity models and !l;-solutions.
In §1 we study the geometry at spatial infinity of n-dimensional (n ~ 3)
noncompact /\;-solutions and in particular we show that they have asymptotic
scalar curvature ratio ASCR = oo and asymptotic volume ratio AVR = 0.
Moreover, we show that the result of having AVR = 0 does not require
the !l;-noncollapsed at all scales assumption. These results on ASCR and
AVR are related to the fact that 3-dimensional noncompact K-solutions are
asymptotically cylindrical at infinity.
In §2 we discuss two results:
(1) solutions which are almost ancient and have bounded nonnegative
curvature operator are collapsed at large scales (Proposition 20.4) and
(2) a curvature estimate in noncollapsed balls (Proposition 20.6).
As we shall see, either of these two results may be used to prove Perelman's
compactness modulo scaling theorem.
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