- PARABOLIC DILATIONS 237
REMARK 8.6. The distinction between ancient and eternal Type II mod-els should be carefully noted. It arises because any limit of a Type II sin-
gularity is noncompact. In order to study its geometry at spatial infinity,
one may take a second limit by a procedure called dimension reduction.
(Dimension reduction will be discussed briefly in Section 4 and thoroughly
in a planned successor to this volume.) This technique produces ancient
solutions which are either Type I or Type II.
EXAMPLE 8.7. The Rosenau solution introduced in Section 3.3 is an
example of a Type II ancient solution that is not eternal but has a 'backward
limit' (as described in Section 6 below) that is an eternal solution. In fact,
this limit is the cigar soliton studied in Section 2.1.
REMARK 8.8. It is finite-time singularities (Type I and Type Ila) that
are most important in applications of the Ricci fl.ow program towards proving
the Geometrization Conjecture for closed 3-manifolds.
EXAMPLE 8.9. In dimension n = 3, the neckpinch (Section 5 of Chapter
2) and the conjectured degenerate neckpinch (Section 6 of Chapter 2) are
the canonical examples of Type I and Type Ila singularities, respectively.- Parabolic dilations
The basic idea of dilating about a finite-time singularity is to choose
a sequence of points and times where the norm of the curvature tends to
infinity and is comparable to its maximum in sufficiently large spatial and
temporal neighborhoods of the chosen points and times. Heuristically, one
trains a microscope at those neighborhoods and magnifies so that the norm
of the curvature is uniformly bounded from above and is equal to · 1 at
the chosen points. Because Perelman's No Local Collapsing theorem [105]
provides an appropriate injectivity estimate in those neighborhoods, one can
then obtain a nontrivial limit solution of the Ricci fl.ow. Complete solutions
which arise as such limits have special properties which suggest that they
can be classified in a way that will yield information about the singularity.
We shall make these ideas precise in what follows. The case of an infinite
time singularity is analogous, and will be considered in Section 5.
3.1. How to choose a sequence of points and times. If (Mn, g (t))
is a solution of the Ricci fl.ow on a maximal time interval 0 :S t < T :S oo, we
may study its properties by dilating the solution about a sequence of points
and times (xi, ti), where Xi E Mn and ti / T. Of course, there are many
possible sequences of points and times around which to dilate, and the limits
we get may depend on our choices. However, each sequence should at the
very least satisfy certain conditions which we outline below. The guiding
principle is that if we want to get a complete smooth limit fl.ow, then we want
the norm of the curvature of each dilated solution to be suitably bounded
in an appropriate parabolic neighborhood of each (.xi, ti)·