Paper · Feb 16, 2023

Mean curvature flow with generic initial data II

Otis Chodosh, Kyeongsu Choi, Felix Schulze

math.DGWider flowsarXiv:2302.08409
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Abstract

We show that the mean curvature flow of a generic closed surface in avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical end cannot arise generically.

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Mean curvature flowSelf-shrinkers & translatorsSingularities

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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 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)6Chu 2023 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)1Chodosh 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)8Hirsch 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)4Chodosh 2023 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)2Chodosh 2024 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)5Ghosh 2026 Formation and structural stability of nondegenerate neckpinches (Kyeongsu Choi, Panagiota Daskalopoulos and Natasa Sesum, 2026)3Choi 2026 Morse resolution of mean curvature flows with cylindrical singularities (Richard H. Bamler, Felix Schulze and Lu Wang, 2026)7Bamler 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 low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows PDF

    Topics Singularities · Same authors Otis Chodosh, Felix Schulze · Shared terms “initial data”, “generic closed”

  2. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows PDF

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

  3. Formation and structural stability of nondegenerate neckpinches

    Kyeongsu Choi, Panagiota Daskalopoulos and Natasa SesumWider flows PDF

    Topics Self-shrinkers & translators, Singularities · Same author Kyeongsu Choi · Shared terms “cylindrical self”, “initial data”

  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. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows PDF

    Topics Self-shrinkers & translators, Singularities · Shared terms “cylindrical self”, “tangent flow”

  6. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Self-shrinkers & translators, Singularities

  7. Morse resolution of mean curvature flows with cylindrical singularities

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

    Topics Singularities · Same author Felix Schulze · Shared terms “tangent flow”, “multiplicity”

  8. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows PDF

    Topics Self-shrinkers & translators, Singularities · Shared terms “cylindrical self”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

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

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.