Paper · Sep 1, 2025

Regularity of cylindrical singular sets of mean curvature flow

Ao Sun, Zhihan Wang, Jinxin Xue

math.DGWider flowsarXiv:2509.01707
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Abstract

In this paper, we study the -cylindrical singular set of mean curvature flow in for each . We prove that they are locally contained in a -dimensional -submanifold after removing some lower-dimensional parts. Moreover, if the -cylindrical singular set is a -submanifold, then its curvature is determined by the asymptotic profile of the flow at these singularities. As a byproduct, we provide a detailed asymptotic profile and graphical radius estimate at these singularities. The proof is based on a new -distance non-concentration property that we introduced in [SWX25], modified into a relative version that allows us to modulo those low eigenmodes that are not decaying fast enough and do not contribute to the curvature of the singular set.

Comments: 61 pages, comments are welcome!

Mean curvature flowSingularities

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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 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)6Sun 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)4Sun 2022 Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy (Ao Sun and Jinxin Xue, 2022)2Sun 2022 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)5Sun 2023 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)7Chu 2023 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)3Sun 2025 Mean curvature flows with prescribed singular sets (Raphael Tsiamis, 2026)8Tsiamis 2026 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)1Sun 2026
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  2. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

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    Topics Singularities · Same authors Ao Sun, Jinxin Xue

  3. Passing through nondegenerate singularities in mean curvature flows

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    Topics Singularities · Same authors Ao Sun, Zhihan Wang, Jinxin Xue

  4. Generic regularity of Level Set Flows with spherical singularity

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    Topics Singularities · Same authors Ao Sun, Jinxin Xue

  5. On mean curvature flow translators with prescribed ends

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    Topics Mean curvature flow · Same authors Ao Sun, Zhihan Wang

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

    Ao Sun and Jinxin Xue PDF

    Topics Singularities · Same authors Ao Sun, Jinxin Xue

  7. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Singularities · Same author Ao Sun

  8. Mean curvature flows with prescribed singular sets

    Raphael TsiamisWider flows PDF

    Topics Mean curvature flow · Shared terms “singular set”

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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)

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