Neckpinch singularity of Ricci flow

Question

I apologise if this question is unclear as I do not know much about the Ricci flow and am only asking out of curiosity. My understanding is that a neckpinch singularity is a local singularity in the sense that it occurs on a compact subset of a manifold. The classic picture is that of a dumbbell manifold, where a local singularity forms after a finite time as the neck of the dumbbell contracts under the Ricci flow.

This is in contrast to an example like the shrinking sphere, which describes a global singularity. My question is whether a neckpinch singularity must necessarily be a local singularity from the formal definition, or if there is some sense in which it is possible to have something like a global neckpinch singularity (at least intuitively).

Edit: I've thought about it again and obviously the neckpinch can only be a type of local singularity. I don't know if there is something vaguely similar which is somehow classed as global, as singularities of Ricci flow is not a subject I know.

Answer 1

It's not entirely clear what you mean by a local singularity, or what it might mean for a local singularity to be "global." I'll give one attempt to make those ideas precise. However, I won't be able to give an answer because the singularity profiles are not fully understood when the dimension is greater than three.

We can say that a Ricci flow singularity is local when its blow-up limit is non-compact. The neckpinch would be the prototypical example as the blow-up is something akin to a cylinder. The fact that the profile of the singularity is noncompact means that there are regions where the curvature is much smaller then the maximum curvature. Here are two possible definitions for a "global" local singularity, either as a singularity where the curvature goes to infinity everywhere (even though it is happening much faster at some points rather than others) or perhaps as a singularity where the diameter of the manifold converges to zero at the singular time. Unfortunately, I don't know whether this is possible and I believe this is an open question.

What I can say is that such singularities can occur for curve shortening flow, which is another geometric flow that was introduced by Hamilton in the 1980s. In particular, if you start with a symmetric curve which is shaped like a figure 8 (i.e. the infinity symbol), under curve shortening flow its length will shrink to zero at the singular time, so in one sense the curve shrinks to a point. However, the blow-up profiles will be two separate grim-reaper curves (depending on which side of the figure 8 you blow up at), which are not compact.