Paper · Jan 18, 2019

Uniqueness of asymptotically conical tangent flows

Otis Chodosh, Felix Schulze

math.DGWider flowsarXiv:1901.06369v2
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Abstract

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for mean curvature flow of hypersurfaces. In particular, this implies that when a mean curvature flow has a multiplicity-one conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to "flow through" such a singularity.

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Submitted
Jan 18, 2019
Last revised
Dec 16, 2020 (v2)
Subjects
math.DG, math.AP
DOI
10.1215/00127094-2020-0098

Comments: 40 pages. Final version, to appear in Duke Mathematical Journal

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & 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 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 Mean curvature flow with generic initial data (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2020)5Chodosh 2020 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)8Khan 2020 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)2Lee 2023 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)3Chodosh 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)1Chodosh 2023 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)6Chodosh 2024 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)4Daniels-Holg 2025 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)7Ghosh 2025
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 from Conical Singularities

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

    Topics Boundaries & non-compact, Singularities · Same authors Otis Chodosh, Felix Schulze · Shared terms “conical singularities”, “asymptotically conical”, “isolated”

  2. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows

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

  3. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

    Topics Self-shrinkers & translators, Singularities · Same authors Otis Chodosh, Felix Schulze · Shared terms “tangent flow”, “multiplicity”

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

    J. M. Daniels-Holgate and Or HershkovitsWider flows

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

  5. Mean curvature flow with generic initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same authors Otis Chodosh, Felix Schulze · Shared terms “asymptotically conical”

  6. Revisiting generic mean curvature flow in

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

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

  7. Cylindrical tangent flows in mean curvature flow

    Sourav GhoshWider flows

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

  8. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

    Ilyas KhanWider flows

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

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) 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)

In the guidebook

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    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.