Paper · Sep 2, 2024

Revisiting generic mean curvature flow in

Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze

math.DGWider flowsarXiv:2409.01463v3
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Abstract

Bamler–Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R^3 and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a short density-drop theorem plus the Bamler–Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken's conjecture – without relying on the strict genus drop theorem for one-sided ancient flows previously established by the authors.

Comments: Final version. To appear in a special bicentennial issue of J. Reine Angew. Math. (Crelle)

Mean curvature flowAncient solutionsSingularities

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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 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)8Lotay 2022 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)1Chodosh 2023 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)2Chodosh 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)4Chodosh 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)5Choi 2024 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)6Choi 2024 Ancient cylindrical flows with fast or slow convergence (Kyeongsu Choi and Robert Haslhofer, 2026)3Choi 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 flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Kyeongsu Choi, Felix Schulze · Shared terms “multiplicity”, “tangent flow”

  2. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Christos Mantoulidis, Felix Schulze

  3. Ancient cylindrical flows with fast or slow convergence

    Kyeongsu Choi and Robert HaslhoferWider flows PDF

    Topics Ancient solutions, Singularities · Same author Kyeongsu Choi · Shared terms “Bamler”

  4. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze

  5. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows PDF

    Topics Ancient solutions · Same author Kyeongsu Choi

  6. Uniqueness of tangent flows at infinity for finite-entropy shortening curves

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows PDF

    Topics Ancient solutions, Singularities · Same author Kyeongsu Choi · Shared terms “tangent flow”

  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”, “multiplicity”

  8. Ancient solutions and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows PDF

    Topics Ancient solutions · Same author Felix Schulze

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.