Stability of neckpinch singularities
Felix Schulze, Natasa Sesum
TL;DR
The paper proves that neckpinch singularities in mean curvature flow are dynamically stable under small perturbations when there are finitely many such singularities at the first singular time, and that nondegenerate neckpinches remain nondegenerate under perturbations. The authors develop a local-to-global stability framework using Brakke flows, limit/tangent flows, and the mean convex neighborhood theorem to constrain possible singularity models near neckpinches, establishing that perturbations preserve neckpinch-type behavior and, in the nondegenerate case, the nature of the singularity. They connect neckpinch stability to the Type I/Type II classification via Hamilton rescaling, showing nondegenerate neckpinches align with Type I, while degenerate ones align with Type II, supporting the conjectured stability of Type I singularities. The results provide a rigorous pathway toward understanding generic singularities in mean curvature flow and have implications for the broader program surrounding Huisken’s conjecture on generic singularities.
Abstract
In this paper, we study the stability of neckpinch singularities. We show that if a mean curvature flow $\{M_t\}$ develops only finitely many neckpinch singularities at the first singular time, then the mean curvature flow starting at any sufficiently small perturbation of $M_0$ can also develop only neckpinch type singularities at the first singular time. We also show stability of nondegenerate neckpinch singularities in the above sense, which speaks in favor of stability of Type I singularities.