Paper · Nov 18, 2020

An unknottedness result for self shrinkers with multiple ends

Alexander Mramor

math.DGWider flowsarXiv:2011.09373v5
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Abstract

In this article we prove an unknottedness result for self shrinkers in with multiple asymptotically conical ends which bound a handlebody in a natural sense, using the mean curvature flow. As a corollary of this and previous work, asymptotically conical self shrinkers with two ends are unknotted.

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Submitted
Nov 18, 2020
Last revised
Jun 29, 2023 (v5)
Subjects
math.DG, math.GT

Comments: Revisited and majorly revised. Results and exposition improved, including some new arguments/details added. Comments welcome!

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compact

Its neighbourhood

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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 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 Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end (Siao-Hao Guo, 2016)7Guo 2016 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)3Bernstein 2017 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)6Chodosh 2019 An unknottedness result for noncompact self shrinkers (Alexander Mramor, 2020)1Mramor 2020 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)2Khan 2020 Noncompact self-shrinkers for mean curvature flow with arbitrary genus (Reto Buzano, Huy The Nguyen and Mario B. Schulz, 2021)4Buzano 2021 Self-shrinkers whose asymptotic cones fatten (Daniel Ketover, 2024)5Ketover 2024 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)8Xie 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An unknottedness result for noncompact self shrinkers

    Alexander Mramor

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same author Alexander Mramor · Shared terms “unknottedness”, “conical ends”, “asymptotically conical”

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

    Ilyas KhanWider flows

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

  3. 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 “conical self”, “asymptotically conical”

  4. Noncompact self-shrinkers for mean curvature flow with arbitrary genus

    Reto Buzano, Huy The Nguyen and Mario B. SchulzWider flows

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

  5. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows

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

  6. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

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

  7. Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

    Siao-Hao GuoWider flows

    Topics Self-shrinkers & translators · Shared terms “conical ends”

  8. 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 “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.

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) 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.