326 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHESThe other necessary task is finding the set Bk and the corresponding sets of
initial data 9[.BJ (to) with properties as described above. As shown in [12], this task
is relatively straightforward, with a strong reliance on the Hermite polynomials and
eigenvalues which arise in the formal matched asymptotic analysis.4. Concluding remarks
The work discussed here determines the asymptotic behavior of a range of
rotationally symmetric Ricci fl.ow solutions which develop degenerate neckpinch
singularities. Should one expect to find the asymptotic behavior discussed here in
generic solutions which develop this sort of singularity? Hamilton has conjectured
that, in the space of all Ricci fl.ow solutions on a closed manifold, degenerate neck-
pinches are nongeneric; see Problem 36.1 in this chapter and seep. 294 in [77]. On
the other hand, do degenerate neckpinch singularities always develop from initial
data sets lying at the threshold between sets leading to nondegenerate neckpinch
singularities and sets which (for volume normalized Ricci fl.ow) do not become sin-
gular? Compare with Problem 8.25 in [77].
The numerical work discussed in Subsection 1.1 indicates that at least among
rotationally symmetric solutions, degenerate neckpinch singularities do generically
develop from threshold data, and they generically show the behavior discussed here.
However, such a result is far from proven.
What about neckpinch singularities forming in the Ricci fl.ow of nonrotationally
symmetric geometries? A consequence of Perelman's work is that all finite-time
singularities on closed 3-manifolds are asymptotically rotationally symmetric after
possibly passing to a finite cover. P erelman proved that 3-dimensional singularity
formation is modeled by canonical neighborhoods. In particular, unless one is on
a spherical space form, one can always extract a round cylinder limit or its Z 2 -
quotient. In the case of 3-dimensional Type II singularities, a Bryant soliton limit
always exists (a proof of this fact also uses Brendle's classification of ,,;-noncollapsed
3-dimensional steady solitons).
In all dimensions it has been conjectured that if the initial geometries are
close to being rotationally symmetric or if they differ from rotationally symmetric
geometries in relatively controlled ways, then the resulting Ricci fl.ow solutions
are likely to become increasingly round and then are likely to exhibit the same
asymptotic behavior as rotationally symmetric solutions. This conjecture has been
explored to a limited extent for nondegenerate neckpinch Ricci fl.ow solutions [155]
and more extensively for nondegenerate neckpinch formation in mean curvature
flows [115], [114]. While these results are very limited and have not yet explored
degenerate neckpinch formation in nonrotationally symmetric Ricci fl.ow, they do
all support the conjecture. This suggests that the study of the asymptotic behavior
of rotationally symmetric Ricci fl.ow solutions in which neckpinch singularities form
could in fact be describing the asymptotic behavior of a much wider range of Ricci
fl.ow neckpinch solutions.