Paper · May 24, 2024

Closed mean curvature flows with asymptotically conical singularities

Tang-Kai Lee, Xinrui Zhao

math.DGarXiv:2405.15577v3
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Abstract

In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Ważewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent–Ilmanen–Velázquez and Chodosh–Daniels-Holgate–Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans–Spruck.

Comments: 23 pages; v2: A new result was added as a corollary thanks to Alec Payne's suggestion; v3: slight modification with references updated

Ricci flowMean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesWeak & singular flowsBoundaries & non-compact

Its neighbourhood

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 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)5Stolarski 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)2Lee 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)3Chodosh 2023 An Intersection Principle for Mean Curvature Flow (Tang-Kai Lee and Alec Payne, 2025)7Lee 2025 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)6Daniels-Holg 2025 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)8Xie 2025 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)1Chen 2026 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)4Braxton 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Closed mean curvature flows with prescribed tangent flows

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

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Same authors Tang-Kai Lee, Xinrui Zhao · Shared terms “closed mean”, “asymptotically conical”

  2. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Boundaries & non-compact, Singularities · Same authors Tang-Kai Lee, Xinrui Zhao · Shared terms “conical singularities”, “asymptotically conical”, “Chodosh”

  3. Mean Curvature Flow from Conical Singularities

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

    Topics Weak & singular flows, Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “set flow”, “level set”

  4. A stable self-shrinking Möbius bundle in

    Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu PDF

    Topics Self-shrinkers & translators, Singularities, Ricci solitons · Same author Tang-Kai Lee

  5. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

    Maxwell Stolarski PDF

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Shared terms “singularities modeled”, “asymptotically conical”

  6. Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

    J. M. Daniels-Holgate and Or HershkovitsWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “asymptotically conical”

  7. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows PDF

    Topics Weak & singular flows, Singularities · Same author Tang-Kai Lee · Shared terms “set flow”, “level set”, “fattening”

  8. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “conical self”, “asymptotically conical”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

In the guidebook

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    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.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.