Paper · Mar 30, 2020

A mountain-pass theorem for asymptotically conical self-expanders

Jacob Bernstein, Lu Wang

math.DGWider flowsarXiv:2003.13857
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Abstract

We develop a min-max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there exists a new asymptotically conical self-expander trapped between the two.

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Submitted
Mar 30, 2020
Subjects
math.DG, math.AP

Comments: 49 pages, 0 figure

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfaces

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)7Fong 2016 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)1Bernstein 2017 Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2018)2Bernstein 2018 Topological Uniqueness for Self-expanders of Small Entropy (Jacob Bernstein and Lu Wang, 2019)3Bernstein 2019 Relative expander entropy in the presence of a two-sided obstacle and applications (Jacob Bernstein and Lu Wang, 2019)5Bernstein 2019 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)8Khan 2020 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)4Xie 2025 The global structure of bi-rotational self-expanders (Gregŕio Silva Neto, 2026)6Neto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The space of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

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

  2. 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 · Same authors Jacob Bernstein, Lu Wang · Shared terms “conical self”, “self expanders”, “asymptotically conical”

  3. Topological Uniqueness for Self-expanders of Small Entropy

    Jacob Bernstein and Lu WangWider flows

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

  4. 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”, “self expanders”, “asymptotically conical”

  5. 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”

  6. The global structure of bi-rotational self-expanders

    Gregŕio Silva NetoWider flows

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

  7. 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”

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

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.