Formation and structural stability of nondegenerate neckpinches
Abstract
In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.
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BibTeX
@misc{choi2026formation,
title = {{Formation and structural stability of nondegenerate neckpinches}},
author = {Kyeongsu Choi and Panagiota Daskalopoulos and Natasa Sesum},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08112},
eprint = {2609.08112},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08112},
}
RIS
TY - UNPB
TI - Formation and structural stability of nondegenerate neckpinches
AU - Kyeongsu Choi
AU - Panagiota Daskalopoulos
AU - Natasa Sesum
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08112
UR - https://arxiv.org/abs/2609.08112
AB - In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.
ER -