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).