I am aware that in general for the Ricci flow, neckpinches are local singularities (ie. they occur on a compact subset of the manifold). The usual picture is that of a manifold shaped like a dumbbell which develops a local singularity as the neck contracts after a finite amount of time. This in contract to the shrinking sphere which describes a global singularity.
Is it also possible in some sense to have a global neckpinch singularity? For example, can one flow the dumbbell until the neck has contracted to a neckpinch and then cut away and discard the two spheres either side of the neck so that a global neckpinch singularity remains?