Paper · Jan 9, 2025

Singularities of mean curvature flow with bounded mean curvature and Morse index

Yongheng Han

math.DGWider flowsarXiv:2501.05489v3
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Abstract

We study the multiplicity of the singularities of mean curvature flow with bounded mean curvature and Morse index. For , we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in .

Comments: 27 pages, 5 figures

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 Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary (Dongyeong Ko, 2023)5Ko 2023 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)2Han 2023 Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature (Zichang Liu, 2024)3Liu 2024 Semi-Continuity of the Morse Index for Ricci Shrinkers (Louis Yudowitz, 2024)6Yudowitz 2024 Well-posedness of mean curvature flow (Yongheng Han, 2026)7Han 2026 Morse resolution of mean curvature flows with cylindrical singularities (Richard H. Bamler, Felix Schulze and Lu Wang, 2026)4Bamler 2026 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)8Daniels-Holg 2026 Singularities of Mean Curvature Flow with Bounded Total Curvature (Yongheng Han and Zhen Wang, 2026)1Han 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 Total Curvature

    Yongheng Han and Zhen WangWider flows PDF

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

  2. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

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

  3. Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

    Zichang LiuWider flows PDF

    Topics Singularities · Shared terms “bounded mean”, “singular time”

  4. Morse resolution of mean curvature flows with cylindrical singularities

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

    Topics Singularities · Shared terms “Morse”, “multiplicity”

  5. Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary

    Dongyeong KoWider flows PDF

    Topics Singularities · Shared terms “Morse index”

  6. Semi-Continuity of the Morse Index for Ricci Shrinkers

    Louis Yudowitz PDF

    Shared terms “Morse index”

  7. Well-posedness of mean curvature flow

    Yongheng HanWider flows PDF

    Topics Mean curvature flow · Same author Yongheng Han

  8. Generic mean curvature flow has finitely many singularities

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

    Topics Singularities · Shared terms “Morse”, “multiplicity”

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)

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

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