Paper · May 15, 2020

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

Ilyas Khan

math.DGWider flowsarXiv:2005.07611v3
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Abstract

Let be an -dimensional cone immersed in . In this paper, we show that if is a properly immersed mean curvature flow self-shrinker which is smoothly asymptotic to , then it is unique and converges to with unit multiplicity. Furthermore, if and are self-expanders that both converge to smoothly asymptotically and their separation decreases faster than in the Hausdorff metric, then the images of and coincide.

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Submitted
May 15, 2020
Last revised
Jun 20, 2023 (v3)
Subjects
math.DG

Comments: 44 pages. Exposition improved, several remarks added, various arguments corrected and expanded

Mean curvature flowSelf-shrinkers & translatorsStability & 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 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)1Fong 2016 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)2Bernstein 2017 Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2018)5Bernstein 2018 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)4Chodosh 2019 A mountain-pass theorem for asymptotically conical self-expanders (Jacob Bernstein and Lu Wang, 2020)6Bernstein 2020 An unknottedness result for self shrinkers with multiple ends (Alexander Mramor, 2020)8Mramor 2020 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)7Daniels-Holg 2025 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)3Xie 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

    Frederick Tsz-Ho Fong and Peter McGrathWider flows

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

  2. The space of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

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

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

    Junming XieWider flows

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

  4. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

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

  5. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

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

  6. A mountain-pass theorem for asymptotically conical self-expanders

    Jacob Bernstein and Lu WangWider flows

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

  7. 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, Stability & uniqueness · Shared terms “asymptotically conical”, “multiplicity”

  8. An unknottedness result for self shrinkers with multiple ends

    Alexander MramorWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “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.

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) 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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.