Paper · Sep 16, 2026

Mean curvature flows with cylindrical singularities II: stability and genericity

Ao Sun, Jinxin Xue

math.DGWider flowsarXiv:2609.19390
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Abstract

In [SX25], we introduced the notion of nondegenerate cylindrical singularity using the ideas from dynamical systems. In this paper, we further study the properties of these singularities, showing that they are stable under small perturbations. Furthermore, we show that we can perturb some mean curvature flow with degenerate singularities to produce nondegenerate singularities.

Comments: 30 Pages. This is the second part of ver 1 of https://arxiv.org/abs/2210.00419v1, with more details. Much progress has been made on this problem since 2022, and it is hard for us to keep track of it all; to reflect the original ideas in this work, we decided to keep our original references

Mean curvature flowSingularitiesStability & uniqueness

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Connected papers

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 2021, 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 Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities (Ao Sun and Jinxin Xue, 2021)4Sun 2021 On the Stability of Cylindrical Singularities of the Mean Curvature Flow (Jingxuan Zhang, 2021)6Zhang 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)5Sun 2022 Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)1Sun 2022 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)8Chu 2023 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)2Sun 2025 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)7Ghosh 2025 Regularity of cylindrical singular sets of mean curvature flow (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)3Sun 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

    Ao Sun and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue · Shared terms “cylindrical singularities”, “nondegenerate cylindrical”

  2. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue · Shared terms “nondegenerate singularities”, “cylindrical singularities”, “nondegenerate cylindrical”

  3. Regularity of cylindrical singular sets of mean curvature flow

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue

  4. Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

    Ao Sun and Jinxin Xue PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue · Shared terms “perturbation”

  5. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue

  6. On the Stability of Cylindrical Singularities of the Mean Curvature Flow

    Jingxuan ZhangWider flows PDF

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

  7. Cylindrical tangent flows in mean curvature flow

    Sourav GhoshWider flows PDF

    Topics Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”

  8. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Singularities · Same author Ao Sun · Shared terms “perturbation”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, 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)

In the guidebook

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    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.