Paper · Nov 21, 2018

On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature

Haozhao Li, Bing Wang

math.DGWider flowsarXiv:1811.08654v2
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Abstract

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-I. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in R^3 is at least of type-I.

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Submitted
Nov 21, 2018
Last revised
Mar 24, 2021 (v2)
Subjects
math.DG

Comments: 93 pages. Final version to appear in Journal of the European Mathematical Society

Mean curvature flowSelf-shrinkers & translatorsSingularitiesLong-time existence & convergence

Its neighbourhood

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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 2016, 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 The extension problem of the mean curvature flow (I) (Haozhao Li and Bing Wang, 2016)1Li 2016 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)3Sun 2018 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)7Chodosh 2019 Mean curvature flow with multiplicity 2 convergence in ℝ³ (Jingwen Chen and Ao Sun, 2023)5Chen 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)6Chen 2024 Singularities of mean curvature flow with bounded mean curvature and Morse index (Yongheng Han, 2025)8Han 2025 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)2Li 2026 Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)4Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The extension problem of the mean curvature flow (I)

    Haozhao Li and Bing WangWider flows

    Topics Mean curvature flow · Same authors Haozhao Li, Bing Wang · Shared terms “embedded mean”, “closed smooth”, “smooth embedded”

  2. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Long-time existence & convergence, Singularities · Same authors Haozhao Li, Bing Wang

  3. Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

    Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “multiplicity”, “Ilmanen”

  4. Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Singularities · Same authors Haozhao Li, Bing Wang

  5. Mean curvature flow with multiplicity 2 convergence in

    Jingwen Chen and Ao SunWider flows

    Topics Long-time existence & convergence · Shared terms “multiplicity”, “smooth embedded”

  6. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows

    Topics Long-time existence & convergence · Shared terms “multiplicity”, “smooth embedded”

  7. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “multiplicity”, “singular time”

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

    Yongheng HanWider flows

    Topics Singularities · Shared terms “embedded mean”, “closed smooth”, “smooth embedded”

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 in the L² sense. 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 can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

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  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.