a neck-pinch singularity
Imagine a dumbbell-shaped surface: two fat round blobs joined by a thin neck. Now run Ricci flow, which shrinks regions of high curvature fastest. The neck is the most sharply curved part, so it shrinks faster than the blobs on either side — and it keeps shrinking, getting thinner and thinner until, at a finite time, it pinches off to a point and the manifold tries to split in two. That pinch is a neck-pinch singularity, the simplest and most important way Ricci flow goes singular, and the prototype for everything surgery has to handle.
Concretely, near a neckpinch the geometry looks, after rescaling, like a shrinking round cylinder S^{n-1} x R — a 'neck' whose cross-sectional sphere collapses while the long direction stays roughly fixed. As the singular time T approaches, the curvature blows up like 1/(T - t) on the neck (a Type I rate) and the metric on the cross-section S^{n-1} shrinks to zero radius. The model solution is the shrinking round sphere times a line, which is a self-similar (gradient shrinking soliton) solution of Ricci flow, and the rigorous statement that genuine neckpinches form on dumbbell metrics was proved by Angenent and Knopf. A degenerate neckpinch is a more delicate variant where one side caps off at the same instant the neck pinches, producing a cusp-like model rather than a clean cylinder.
Neckpinches matter because they are exactly the singularities Hamilton's and Perelman's surgery is designed to cut out and repair: you wait until a neck is thin enough that its geometry is provably close to the standard cylinder model, then you cut along the neck and glue in two smooth round caps, continuing the flow past the singular time. The honest subtlety is that not every Ricci-flow singularity is a neckpinch — there are more complicated singularity models — but Perelman's canonical-neighborhood theorem shows that in dimension three every high-curvature region is, up to rescaling, close to a neck, a cap, or a closed model, which is precisely what makes three-dimensional surgery possible. Do not assume the dumbbell picture captures all singular behaviour in higher dimensions, where the singularity zoo is genuinely larger.
Take a barbell metric on S^3: two large round 3-balls joined by a long thin S^2 x interval neck. Ricci flow shrinks the neck's S^2 cross-section toward zero while the two balls stay large, and at the singular time the curvature on the neck diverges like 1/(T - t) — exactly the shrinking-cylinder S^2 x R model.
A dumbbell on S^3: the neck pinches off as a shrinking S^2 x R cylinder, the canonical Ricci-flow singularity.
A neckpinch is a Type I (curvature ~ 1/(T-t)) singularity modeled on a gradient shrinking soliton; degenerate neckpinches and Type II singularities (faster blow-up, e.g. modeled on the Bryant soliton) also occur, so 'neckpinch' should not be used as a synonym for 'Ricci-flow singularity' in general.