Paper · Jan 26, 2023

Uniqueness of conical singularities for mean curvature flows

Tang-Kai Lee, Xinrui Zhao

math.DGWider flowsarXiv:2301.10883v2
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Abstract

In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.

Comments: 19 pages

Mean curvature flowSingularitiesStability & uniquenessBoundaries & non-compact

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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 Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities (Ao Sun and Jinxin Xue, 2021)6Sun 2021 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)4Chodosh 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)1Lee 2024 Arnold-Thom conjecture for the arrival time of surfaces (Tang-Kai Lee and Jingze Zhu, 2024)7Lee 2024 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)3Daniels-Holg 2025 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)5Ghosh 2025 On the rate of convergence of cylindrical singularity in mean curvature flow (Yiqi Huang and Xinrui Zhao, 2025)8Huang 2025 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)2Chen 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 asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao PDF

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

  2. Closed mean curvature flows with prescribed tangent flows

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

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

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

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

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Shared terms “conical singularities”, “asymptotically conical”

  4. Mean Curvature Flow from Conical Singularities

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

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

  5. Cylindrical tangent flows in mean curvature flow

    Sourav GhoshWider flows PDF

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Shared terms “tangent flow”

  6. Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

    Ao Sun and Jinxin Xue PDF

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

  7. Arnold-Thom conjecture for the arrival time of surfaces

    Tang-Kai Lee and Jingze ZhuWider flows PDF

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

  8. On the rate of convergence of cylindrical singularity in mean curvature flow

    Yiqi Huang and Xinrui ZhaoWider flows PDF

    Topics Singularities, Stability & uniqueness · Same author Xinrui Zhao

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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