Paper · Oct 13, 2026

Singularities of Mean Curvature Flow with Bounded Total Curvature

Yongheng Han, Zhen Wang

math.DGWider flowsarXiv:2610.18197
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Abstract

We prove that every tangent flow at the first singular time of a smooth, closed, embedded mean curvature flow has smooth support, provided that the total curvature satisfies . This extends Ilmanen's smoothness result for tangent flows to higher dimensions.

Comments: 51 pages

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 A local estimate for the mean curvature flow (Zhen Wang, 2021)5Wang 2021 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)3Han 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)4Choi 2024 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)7Ilmanen 2024 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)2Su 2024 Singularities of mean curvature flow with bounded mean curvature and Morse index (Yongheng Han, 2025)1Han 2025 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)8Ghosh 2025 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)6Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of mean curvature flow with bounded mean curvature and Morse index

    Yongheng HanWider flows PDF

    Topics Singularities · Same author Yongheng Han · Shared terms “embedded mean”, “singular time”

  2. On bounds of entropy and total curvature for ancient curve shortening flows

    Wei-Bo Su and Kai-Wei ZhaoWider flows PDF

    Topics Singularities · Shared terms “total curvature”, “tangent flow”

  3. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

    Topics Mean curvature flow · Same author Yongheng Han · Shared terms “total curvature”

  4. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows PDF

    Topics Mean curvature flow · Shared terms “total curvature”

  5. A local estimate for the mean curvature flow

    Zhen WangWider flows PDF

    Topics Singularities · Same author Zhen Wang

  6. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al. PDF

    Topics Singularities · Shared terms “tangent flow”, “embedded mean”

  7. Fattening in mean curvature flow

    Tom Ilmanen and Brian White PDF

    Topics Singularities · Shared terms “singular time”, “tangent flow”

  8. Cylindrical tangent flows in mean curvature flow

    Sourav GhoshWider flows PDF

    Topics Singularities · Shared terms “tangent flow”

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)

In the guidebook

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