[Paper Review] Stability of neckpinch singularities

Felix Schulze, Nataša Šešum|arXiv (Cornell University)|Jun 10, 2020

Geometric Analysis and Curvature Flows8 references4 citations

TL;DR

This paper establishes the stability of nondegenerate neckpinch singularities in mean curvature flow by proving that small C² perturbations of an initial hypersurface that develops only finitely many isolated nondegenerate neckpinch singularities at the first singular time will also develop only nondegenerate neckpinch singularities. The result relies on the mean convex neighborhood theorem and blow-up analysis, showing that such singularities are robust under perturbation and are always of Type I, supporting 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.

Motivation & Objective

• To investigate the stability of neckpinch singularities in mean curvature flow under small perturbations of the initial hypersurface.
• To determine whether nondegenerate neckpinch singularities persist under small C² perturbations of the initial data.
• To confirm that such singularities are of Type I and not Type II, supporting their generic occurrence in mean curvature flow.
• To provide a theoretical foundation for Huisken's conjecture that generic mean curvature flows develop only spherical and cylindrical singularities.

Proposed method

• The authors use the mean convex neighborhood theorem by Choi, Haslhofer, Hershkovits, and White to control the local geometry near potential singularities.
• They apply parabolic rescaling and blow-up analysis to study tangent flows and limit flows at singular points.
• The proof relies on contradiction arguments involving the classification of ancient solutions, particularly ruling out translating bowl solitons.
• They establish that nondegenerate neckpinch singularities satisfy a local Type I bound, which restricts the possible blow-up limits to round cylinders.
• The analysis distinguishes cases based on the asymptotic behavior of rescaling factors and second fundamental form norms.
• The stability result is derived by showing that the absence of degenerate limit flows under small perturbations preserves the nondegenerate neckpinch structure.

Experimental results

Research questions

• RQ1 Does a small C² perturbation of an initial hypersurface that develops only nondegenerate neckpinch singularities at the first singular time also result in only nondegenerate neckpinch singularities?
• RQ2 Can nondegenerate neckpinch singularities be ruled out from developing into degenerate or Type II singularities under perturbation?
• RQ3 Is the local geometry near a nondegenerate neckpinch stable under parabolic rescaling and blow-up analysis?
• RQ4 Do nondegenerate neckpinch singularities satisfy a local Type I curvature bound, and what does this imply about their classification?
• RQ5 Can the classification of ancient solutions (e.g., translating bowls) be used to rule out nondegenerate neckpinch singularities in the limit?

Key findings

• Small C² perturbations of an initial hypersurface that develops only finitely many isolated nondegenerate neckpinch singularities at the first singular time will also develop only nondegenerate neckpinch singularities.
• Nondegenerate neckpinch singularities are always of Type I, as they cannot arise from blow-up limits involving translating bowl solitons.
• The proof relies on the mean convex neighborhood theorem and the classification of ancient solutions under uniform 2-convexity and α-noncollapsing.
• The absence of degenerate limit flows under perturbation ensures that the singularity model remains a multiplicity-one round cylinder.
• The stability result confirms that nondegenerate neckpinches are robust and not expected to occur only non-generically.
• The work supports Huisken’s conjecture by showing that such singularities are stable and thus likely to appear generically in mean curvature flow.