Paper · May 29, 2024

Arnold-Thom conjecture for the arrival time of surfaces

Tang-Kai Lee, Jingze Zhu

math.DGWider flowsarXiv:2405.19064v2
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Abstract

Following Łojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove Łojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in with neck or non-degenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.

Comments: final version, accepted to Duke Math J

Mean curvature flowSingularitiesStability & uniqueness

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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 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)3Lee 2023 Rigidity of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Jingze Zhu, 2025)6Choi 2025 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)4Lee 2025 An Intersection Principle for Mean Curvature Flow (Tang-Kai Lee and Alec Payne, 2025)7Lee 2025 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)8Chen 2026 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)5Braxton 2026 General regularity obstructions for the arrival time equation (Yiqi Huang and Jingze Zhu, 2026)1Huang 2026 Optimal regularity for the arrival time equation (Yiqi Huang and Jingze Zhu, 2026)2Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. General regularity obstructions for the arrival time equation

    Yiqi Huang and Jingze ZhuWider flows PDF

    Topics Singularities · Same author Jingze Zhu · Shared terms “arrival time”, “time function”, “degenerate”

  2. Optimal regularity for the arrival time equation

    Yiqi Huang and Jingze ZhuWider flows PDF

    Topics Mean curvature flow · Same author Jingze Zhu · Shared terms “arrival time”, “time function”, “convex mean”

  3. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Singularities, Stability & uniqueness · Same author Tang-Kai Lee

  4. Planarity and convexity for pinched ancient solutions of mean curvature flow

    Tang-Kai Lee, Keaton Naff and Jingze Zhu PDF

    Topics Mean curvature flow · Same authors Tang-Kai Lee, Jingze Zhu

  5. A stable self-shrinking Möbius bundle in

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

    Topics Singularities, Stability & uniqueness · Same author Tang-Kai Lee

  6. Rigidity of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Jingze ZhuWider flows PDF

    Topics Singularities, Stability & uniqueness · Same author Jingze Zhu

  7. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows PDF

    Topics Singularities · Same author Tang-Kai Lee

  8. Closed mean curvature flows with prescribed tangent flows

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

    Topics Singularities · Same author Tang-Kai Lee

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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