- THE NECKPINCH 39
FIGURE 3. The shrinking cylinder solitonA neckpinch is a special type of local singularity. There exist quanti-
tative measures of necklike behavior for a solution of the Ricci flow. (See
[65], as well as Section 4 of Chapter 9.) However, the following qualitative
characterization will suffice for our present purposes. One says a solution
(Mn+ I, g (t)) of the Ricci flow encounters a neckpinch singularity at time
T < oo if there exists a time-dependent open subset Nt <;;;; Mn+l such that
Nt is diffeomorphic to a quotient of JR x sn by a finite group acting freely,
and such that the pullback of the metric g (t) INt to JR x sn asymptotically
approaches the 'shrinking cylinder' soliton
(2.41) ds^2 + 2 (n - 1) (T - t) 9can
in a suitable sense as t / T. (Here 9can denotes the round metric of radius
1 on sn.)
From the perspective of topology, the neckpinch is perhaps the most
important and intensively studied singularity which the Ricci flow can en-
counter, especially in dimensions three and four. In Chapter 9, we shall see
some reasons why this is so. For example, it is expected that one can per-
form a geometric-topological surgery on the underlying manifold just prior
to a neckpinch in such a way that the maximum curvature of the solution is
reduced by an amount large enough to permit the flow to be continued on
the piece or pieces that remain after the surgery. (We plan to discuss such
surgeries in a successor to this volume.)
From the perspective of asymptotic analysis, on the other hand, remark-
ably little is known about singularity formation in the Ricci flow. For exam-
ple, one cannot find in the literature examples of either formal or rigorous
asymptotic analysis for Ricci flow singularities comparable to what has been
done for the mean curvature flow [7]. (Also see [l].) By contrast, excellent
examples of singularity analysis for related reaction-diffusion equations like
Ut = ~u + uP can be found in [46, 47, 48] and [41], among other sources.
Because neckpinches are such important singularities, we shall study
them in considerable detail in this section. The first rigorous examples of
neckpinches for the Ricci flow were constructed by Sigurd Angenent and
the second author in [6]. In fact, these are the first examples of any sort
of pinching behavior of the Ricci flow on compact manifolds. This section
closely follows that paper. The proofs are considerably more involved than