Paper · Jun 19, 2019

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

Jacob Bernstein, Lu Wang

math.DGarXiv:1906.07863v2
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Abstract

We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same cone. This allows us to begin to develop the variational theory for the relative entropy functional for the associated obstacle problem. We also obtain a version of the forward monotonicity formula for mean curvature flow proposed by Ilmanen.

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Submitted
Jun 19, 2019
Last revised
Mar 30, 2020 (v2)
Subjects
math.DG, math.AP

Comments: 34 pages, 0 figures

Mean curvature flowRicci solitonsSelf-shrinkers & translators

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 Asymptotic structure of almost eigenfunctions of drift Laplacians on conical ends (Jacob Bernstein, 2017)7Bernstein 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)4Bernstein 2018 Generic uniqueness of expanders with vanishing relative entropy (Alix Deruelle and Felix Schulze, 2018)6Deruelle 2018 Topological Uniqueness for Self-expanders of Small Entropy (Jacob Bernstein and Lu Wang, 2019)1Bernstein 2019 A mountain-pass theorem for asymptotically conical self-expanders (Jacob Bernstein and Lu Wang, 2020)3Bernstein 2020 Relative Entropy of Hypersurfaces in Hyperbolic Space (Junfu Yao, 2022)8Yao 2022 On knotted self expanders of the mean curvature flow in R³ (Alexander Mramor and Shengwen Wang, 2026)5Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”

  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. 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”

  5. On knotted self expanders of the mean curvature flow in

    Alexander Mramor and Shengwen Wang

    Topics Self-shrinkers & translators, Ricci solitons · Shared terms “self expanders”

  6. Generic uniqueness of expanders with vanishing relative entropy

    Alix Deruelle and Felix Schulze

    Topics Ricci solitons · Shared terms “relative entropy”, “expanders”

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

    Jacob BernsteinWider flows

    Topics Self-shrinkers & translators · Same author Jacob Bernstein · Shared terms “self expanders”

  8. Relative Entropy of Hypersurfaces in Hyperbolic Space

    Junfu YaoWider flows

    Topics Mean curvature flow · Shared terms “relative entropy”, “monotonicity formula”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 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.