Paper · Feb 7, 2019

Topological Uniqueness for Self-expanders of Small Entropy

Jacob Bernstein, Lu Wang

math.DGWider flowsarXiv:1902.02642v2
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Abstract

For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.

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Submitted
Feb 7, 2019
Last revised
Mar 30, 2020 (v2)
Subjects
math.DG, math.AP

Comments: 33 pages, 0 fiigures

Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness

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 Spectral properties and rigidity for self-expanding solutions of the mean curvature flows (Xu Cheng and Detang Zhou, 2017)6Cheng 2017 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)1Bernstein 2018 Relative expander entropy in the presence of a two-sided obstacle and applications (Jacob Bernstein and Lu Wang, 2019)4Bernstein 2019 A mountain-pass theorem for asymptotically conical self-expanders (Jacob Bernstein and Lu Wang, 2020)3Bernstein 2020 Self-expanders of the mean curvature flow (Knut Smoczyk, 2020)7Smoczyk 2020 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)8Khan 2020 On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows (Letian Chen, 2022)5Chen 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators · Same authors Jacob Bernstein, Lu Wang · Shared terms “self expanders”, “self expanding”, “expanding solutions”

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

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators · Same authors Jacob Bernstein, Lu Wang · Shared terms “self expanders”

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

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators · Same authors Jacob Bernstein, Lu Wang · Shared terms “self expanders”

  4. Relative expander entropy in the presence of a two-sided obstacle and applications

    Jacob Bernstein and Lu Wang

    Topics Self-shrinkers & translators · Same authors Jacob Bernstein, Lu Wang · Shared terms “self expanders”

  5. On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows

    Letian ChenWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “small entropy”, “self expanders”

  6. Spectral properties and rigidity for self-expanding solutions of the mean curvature flows

    Xu Cheng and Detang ZhouWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”, “self expanding”, “expanding solutions”

  7. Self-expanders of the mean curvature flow

    Knut SmoczykWider flows

    Topics Self-shrinkers & translators · Shared terms “self expanders”, “self expanding”, “expanding solutions”

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

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “self expanders”

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)

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.