Chapter 36. Type II Singularities and Degenerate Neckpinches
One of t he key features of Hamilton's scenario for the Ricci flow of 3-dimensional
geometries is the role of neckpinch singularities. The claim is that if a 3-dimensional
Ricci flow solution develops a local singularity, it should generally take the form of
a constricting neckpinch (see §3 of [138] or Subsection 2.3 of Chapter 8 of [77]).
Roughly speaking, a neckpinch is characterized by the local topology 52 x I (for an
interval I) , by the curvature achieving a local maximum at the center of the interval,
and by the curvature in t h e 52 direction dominating that in the I direction. If the
neckpinch is sufficiently constricted, in the sense that t he 52 radii are sufficiently
small , with correspondingly large curvature, then under Ricc i fl.ow the neckpinch
must further constrict and become singular in finite t ime.
In the process of proving the Poincare and geometrization conjectures, Perel-
man's work [312], [313] co nfirms Hamilton's general conjectures regarding this cen-
tral role of neckpinches in the formation of singularities in 3-dimensional Ricci fl.ow.
Although Hamilton's and especially P erelma n 's works say a tremendous amount
about neckpinch singularities, one can further study the detailed nature of these
singularities and t heir asymptotic form.
At least for rotationally symmetric geometries , work by Angenent together with
one of the authors (see [13] or Appendix A in Part I) analyzes the details of the
asymptotic behavior of neckpinch singularities. Their work also shows, as expected,
that (rotationally symmetric) neckpinch singularities are Type I (in the sense that
I Rm l(T-t), for T the time of singularity formation, is uniformly bounded). In other
work by two of the authors and Sesum [155], support is found for the conjecture
t hat the asymptotic behavior of nonrotationally symmetric neckpinch singularities
in Ricci fl.ow is accurat ely modeled by that of rotationally symmetric neckpinches.
Like the neckpinch singularities, the Ricci fl.ow singularit ies which develop from
the extinction of a round 3-sphere, from t he collapse of an 53 whose geometry has
positive Ricci curvature, and from the extinction of a round product 52 x 51 are
all Type I. How might Type II singularit ies develop? One possibility which h as
been proposed by Hamilton is t he following: Consider a family of 3-dimensional
geometries (say, on 53 ) , each with a neckpinch centered at the equator and with the
degree of pinching parametrized (continuously) by A E (0, oo). For very small values
of .A, the pinching is tightly constricted and the Ricci fl.ow becomes singula r in finite
time, while for larger values of A t he constricting is very loose, the scalar curvature
is positive, and t h e (normali zed) Ricci fl.ow converges to the round sphere. Since a
neckpinch singularity forms for small A and does not for large .A, what happens for
intermediate values? Is there a transitional "threshold" or "critical" value Ac for
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