318 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
with the formation of a neckpinch singularity under mean curvature flow. Very con-
sistently, it is found [118] that in a neighborhood of where the neckpinch occurs,
the embedding becomes rounder and rounder, and the features of the developing
neckpinch are essentially identical to those seen in a mean curvature neckpinch
singularity which is initially and always rotationally symmetric. Experience sug-
gests that similar behavior is likely to be seen in similar families of nonrotationally
symmetric Ricci flow degenerate neckpinch singularities.
No matter how emphatically numerical studies might suggest that solutions of
Ricci flow have a certain behavior, such studies do not prove that this is indeed
the case. On the other hand, numerical studies can serve as a strong guide toward
how best to go about proving that this behavior is present. The next two sections
outline a successful program for proceeding from detailed numerical simulations to
a verification that degenerate neckpinch singularities with the numerically observed
behavior indeed occur in 3-dimensional Ricci flow.
2. Matched asymptotic studies of degenerate neckpinches
The goal of a formal matched asymptotics study of the conjectured behavior
of a class of solutions of a given PDE system is to produce a set of approximate
solutions which exhibit that conjectured behavior. To do this, one first formulates a
detailed ansatz which encapsulates most of the features of the conjectured behavior.
Based on this ansatz, one simplifies the PDE system (e.g., via asymptotic expansions
and linearization) and then obtains solutions to the simplified system. One then
checks these solutions for consistency with the ansatz, removing those which fail
this consistency test. Those remaining are the formal (approximate) solutions. By
construction, they should match the conjectured b ehavior.
In most cases, including the study of degenerate neckpinch singularities in [11 J
which we describe here, the ansatz is best defined and applied by cutting the relevant
domain into a set of regions and initially working in each region independently.
The simplified PDE systems are generally very different from one region to another;
hence a crucial part of the consistency check is to match the approximate (regionally
defined) solutions at the regional boundaries. We see this in Subsection 2.7 below.
2.1. Degenerate neckpinch ansatz.
The full ansatz used in [11] for constructing formal approximate Ricci flow
solutions with degenerate neckpinch singularities is best stated in stages. The
first stage- the Primary Ansatz- can be stated without reference to a choice of
coordinates or to the four local regions of study. We state this here. The remaining
stages of the full ansatz depend on coordinate choices and pertain to the individual
separate regions. These are stated below in Subsections 2.3, 2.4, 2.5, and 2.6, in
the context of the regional analyses.
The Primary Ansatz consists primarily of conditions which the initial geome-
tries (Mn+l, go) of the Ricci flows of interest (Mn+l, g(t)) must satisfy. One veri-
fies that most of these conditions are preserved throughout the flow. The Primary
Ansatz also includes a condition that the flow becomes singular in finite time.
ANSATZ 36.2 (Primary Ansatz). The manifold Mn+l is the sphere sn+l
(for n ::=:: 2), and the initial geometries (sn+^1 ,g 0 ) are rotationally symmetric in the
sense that they admit an SO(n + 1) isometry group. (This isometry group action
defines a pair of poles in sn+l and a foliation of 5n+l by sn sections orthogonal
