308 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
the parameter, and what is the asymptotic behavior of the Ricci fl.ow starting at
such a critical geometry?
Rigorous studies by Angenent and Velazquez along with intuitive argument
(see [15]) of analogous mean curvature flows have suggested that the Ricci flows
of critical geometries might develop Type Ila singularit ies (finite-time singularity,
I Rm l(T - t) unbounded, singularity model an eternal solution) and further that
these singularities would be marked by the concentration of curvature at the poles
rather than at the neckpinch, with the fl.ow at the pole modeled by the (steady)
Bryant soliton [37]. Such behavior has been labeled a degenerate neckpinch singu-
larity.
Strong support for this conjecture has been obtained from two programs of
research. The first, that of Gu and Zhu [127], relies on a number of results from
P erelman's work [312], [313] to show that indeed there do exist (rotationally sym-
metric) Ricci fl.ow solutions which develop degenerate neckpinch singularities. They
prove that such solutions exist on 5n+i for all n 2:: 3; however they provide no de-
tails regarding the nature of these singularities apart from showing that t hey are
Type Ila and that the curvature does concentrate at one or both of the poles, where
the singularity is modeled by a Bryant soliton.
The second program of research, carried out by two of the authors and t heir
collaborators, is focussed more on the detailed nature of the fl.ow geometry in the
neighborhood of degenerate neckpinch singularities which form in Ricci fl.ow. This
program, which we survey in this chapter, has been carried out in three stages. The
first stage involves numerical simulations of the Ricci fl.ow of rotationally symmetric
critical^1 geometries. This work, done by Garfinkle and one of the authors [116],
[117] very strongly and clearly indicates that for a wide variety of families of initial
geometries with parametrized degrees of neckpinching, the Ricci fl.ow of the critical
initial geometry in that family forms a degenerate neckpinch singularity. Restricted
for numerical convenience to 5 n+l, these simulations provide models of the behavior
of the geometry near the developing singularity. We describe this work in §1,
where we present some of the general considerations which arise in using numerical
simulations as a tool for studying Ricci fl.ow, along with the details of this particular
numerical study.
The second stage of this program is a formal matched asymptotics study of
the formation of degenerate neckpinch singularities under Ricci fl.ow. As we discuss
in §2, this work (done by Angenent and two of the authors [11]) involves the
construction of a class of very detailed approximate solutions for the fl.ow in a
neighborhood of one of the poles, with all of these approximations (done on 5n+l,
with rotational symmetry) exhibiting degenerate neckpinch behavior. A formal
study of this nature does not prove that Ricci fl.ow solutions which form degenerate
neckpinch singularities and have the prescribed near-the-pole behavior do exist, but
it does strongly suggest that this is the case.
The proof that such solutions do indeed exist is detailed in [12] and is dis-
cussed here in §3. The key idea is to use a modified barrier-type argument to show
that for every approximate solution constructed using formal matched asymptotics,
there is at least one solution of Ricci fl.ow which, near the poles, converges to that
(^1) Here we use the word "critical" in the sense described above for a .>..-parametrized family of
initial geometries.