Paper · Oct 2, 2022

Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

Ao Sun, Jinxin Xue

math.DGWider flowsarXiv:2210.00419v3
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Abstract

This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a ball of radius , and a normal form of the asymptotics. Using the normal form, we can define the nondegeneracy of cylindrical singularities, and we show that nondegenerate cylindrical singularities are isolated in space, have a mean convex neighborhood, and are type-I.

Comments: 43 pages. We divide the original version into two parts, and this is the first part. The second part will be posted in the future. There are more details included, and for the references of later work, some new statements and arguments are added. Comments are welcome!

Mean curvature flowSingularities

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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)8Zhang 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)5Sun 2022 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)2Sun 2025 Regularity of cylindrical singular sets of mean curvature flow (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)3Sun 2025 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)1Sun 2026 Morse resolution of mean curvature flows with cylindrical singularities (Richard H. Bamler, Felix Schulze and Lu Wang, 2026)6Bamler 2026 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)7Daniels-Holg 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 flows with cylindrical singularities II: stability and genericity

    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 “cylindrical singularities”, “nondegenerate cylindrical”, “isolated”

  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 “rescaled mean”

  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. Morse resolution of mean curvature flows with cylindrical singularities

    Richard H. Bamler, Felix Schulze and Lu WangWider flows PDF

    Topics Singularities · Shared terms “cylindrical singularities”, “nondegeneracy”, “mean convex”

  7. Generic mean curvature flow has finitely many singularities

    J. M. Daniels-Holgate, Or Hershkovits and Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Shared terms “generic mean”, “cylindrical singularities”, “mean convex”

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

    Jingxuan ZhangWider flows PDF

    Topics Singularities · Shared terms “cylindrical singularities”, “rescaled mean”, “asymptotics”

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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)

In the guidebook

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