Paper · Jun 10, 2020

Stability of neckpinch singularities

Felix Schulze, Natasa Sesum

math.DGWider flowsarXiv:2006.06118v2
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Abstract

In this paper, we study the stability of neckpinch singularities. We show that if a mean curvature flow develops only finitely many neckpinch singularities at the first singular time, then the mean curvature flow starting at any sufficiently small perturbation of 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.

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Submitted
Jun 10, 2020
Last revised
Apr 25, 2025 (v2)
Subjects
math.DG, math.AP

Comments: 14 pages. Final version. To appear in Comm. Anal. Geom

Mean curvature flowSingularitiesStability & uniqueness

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Neckpinch singularities in fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2016)3Cinti 2016 Analysis of Velacuteazquez's solution to the mean curvature flow with a type II singularity (Siao-Hao Guo and Natasa Sesum, 2017)5Guo 2017 Consequences of strong stability of minimal submanifolds (Jason D. Lotay and Felix Schulze, 2018)6Lotay 2018 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)4Chodosh 2019 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)7Carson 2020 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)8Garfinkle 2022 Mean curvature flow near a peanut solution (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2025)1Angenent 2025 Formation and structural stability of nondegenerate neckpinches (Kyeongsu Choi, Panagiota Daskalopoulos and Natasa Sesum, 2026)2Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow near a peanut solution

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities, Stability & uniqueness · Same author Natasa Sesum · Shared terms “neckpinch singularities”, “nondegenerate neckpinch”, “flow starting”

  2. Formation and structural stability of nondegenerate neckpinches

    Kyeongsu Choi, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities, Stability & uniqueness · Same author Natasa Sesum · Shared terms “neckpinch singularities”, “nondegenerate”, “sufficiently”

  3. Neckpinch singularities in fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

    Topics Singularities · Shared terms “neckpinch singularities”

  4. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Singularities, Stability & uniqueness · Same author Felix Schulze · Shared terms “singular time”

  5. Analysis of Velzquez's solution to the mean curvature flow with a type singularity

    Siao-Hao Guo and Natasa SesumWider flows

    Topics Singularities · Same author Natasa Sesum · Shared terms “perturbation”

  6. Consequences of strong stability of minimal submanifolds

    Jason D. Lotay and Felix SchulzeWider flows

    Topics Singularities, Stability & uniqueness · Same author Felix Schulze · Shared terms “perturbation”

  7. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

    Topics Singularities, Stability & uniqueness · Same author Natasa Sesum

  8. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Singularities, Stability & uniqueness · Shared terms “neckpinch singularities”, “perturbation”, “sufficiently”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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