- THE DEGENERATE NECKPINCH 63
FIGURE 5. A lopsided dumbell shrinking to a round pointAs o: increases from 0 to 1, we may thus imagine that the initial metrics
g°' (0) are dumbbells in which the necks become shorter and fatter while one
of the hemispheres becomes progressively smaller. Such metrics can in fact
be constructed having positive scalar curvature. Then for each o: E [O, 1],
the solution will exist up to a time Ta. < oo when a singularity forms.
(By Theorem 6.45, a finite-time singularity occurs at some time T if and
only if the curvature becomes unbounded as t / T. And by Lemma 6.53,
a finite-time singularity is inevitable if the scalar curvature ever becomes
everywhere positive.) By the continuous dependence of a well-posed PDE on
its initial conditions, one expects that there is some parameter & E (0, 1)
such that the neck pinches off for all o: E [O, &). On the other hand, one
expect for all o: E ( &, 1] that the necks will not pinch off but that the metrics
will approach constant curvature while shrinking to a point. (Depending on
the initial family of metrics g°' (0), there may of course be more than one
such bifurcation point in (0, 1) at which the solutions change qualitatively.)
For the solution go. (t), we expect that the neck pinches off at To. < oo
exactly at the same time that the smaller hemisphere shrinks to a point. As
illustrated in Figure 6, this should result in a cusplike singularity known at
a degenerate neckpinch. (Another discussion of the intuition behind this
example may be found in Section 3 of [63].)
The degenerate neckpinch is the standard conjectural model for what
is known as a slowly forming singularity. As we shall see in Chapter 8, a
finite-time singularity of a solution (Mn, g ( t)) to the Ricci fl.ow is classified
as Type I and called 'rapidly forming' if it occurs at the natural parabolic
growth rate suggested by the example of the round sphere in Subsection 3.1,
hence if there exists C < oo such thatsup IRml · (T - t) :::; C < oo.
Mnx[O,T)