Paper · Apr 24, 2018

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

Jacob Bernstein, Lu Wang

math.DGWider flowsarXiv:1804.09076v2
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Abstract

We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expanders of low entropy and for the set of mean convex self-expanders with strictly mean convex asymptotic cones. From this we deduce that the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones is proper for these classes.

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Submitted
Apr 24, 2018
Last revised
Jul 21, 2018 (v2)
Subjects
math.DG, math.AP

Comments: 19 pages, 0 figures, fix minor typos and update references

Mean curvature flowSelf-shrinkers & translatorsSurfacesLimit spaces & RCDBoundaries & 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 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)6Fong 2016 Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends (Jacob Bernstein, 2017)8Bernstein 2017 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)1Bernstein 2017 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 A mountain-pass theorem for asymptotically conical self-expanders (Jacob Bernstein and Lu Wang, 2020)2Bernstein 2020 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)7Khan 2020 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)4Xie 2025
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. A mountain-pass theorem for asymptotically conical self-expanders

    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”, “self expanding”, “expanding solutions”

  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. 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”, “convex self”

  7. 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”

  8. Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends

    Jacob BernsteinWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same author Jacob Bernstein · 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.