Paper · Jul 20, 2020

Colding Minicozzi Entropy in Hyperbolic Space

Jacob Bernstein

math.DGWider flowsarXiv:2007.10218
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Abstract

This note introduces a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi for submanifolds of Euclidean space. Several properties are proved for this quantity including monotonicity along mean curvature flow in low dimensions and a connection with the conformal volume.

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Submitted
Jul 20, 2020
Subjects
math.DG, math.AP

Comments: 16 pages

Mean curvature flow

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 On the entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)5Zhu 2016 Topological Uniqueness for Self-expanders of Small Entropy (Jacob Bernstein and Lu Wang, 2019)7Bernstein 2019 Relative expander entropy in the presence of a two-sided obstacle and applications (Jacob Bernstein and Lu Wang, 2019)8Bernstein 2019 Relative Entropy of Hypersurfaces in Hyperbolic Space (Junfu Yao, 2022)6Yao 2022 Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds (Jacob Bernstein and Arunima Bhattacharya, 2022)3Bernstein 2022 Rigidity Properties of Colding-Minicozzi Entropies (Jacob Bernstein, 2023)1Bernstein 2023 Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers (Jacob Bernstein, 2024)2Bernstein 2024 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)4Ghosh 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity Properties of Colding-Minicozzi Entropies

    Jacob BernsteinWider flows

    Topics Mean curvature flow · Same author Jacob Bernstein · Shared terms “Minicozzi entropy”, “Colding Minicozzi”, “along mean”

  2. Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers

    Jacob BernsteinWider flows

    Topics Mean curvature flow · Same author Jacob Bernstein · Shared terms “Minicozzi entropy”, “Colding Minicozzi”, “conformal volume”

  3. Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds

    Jacob Bernstein and Arunima BhattacharyaWider flows

    Topics Mean curvature flow · Same author Jacob Bernstein · Shared terms “Minicozzi entropy”, “Colding Minicozzi”, “Euclidean space”

  4. Cylindrical tangent flows in mean curvature flow

    Sourav GhoshWider flows

    Topics Mean curvature flow · Shared terms “Colding Minicozzi”

  5. On the entropy of closed hypersurfaces and singular self-shrinkers

    Jonathan J. ZhuWider flows

    Topics Mean curvature flow · Shared terms “Colding Minicozzi”

  6. Relative Entropy of Hypersurfaces in Hyperbolic Space

    Junfu YaoWider flows

    Topics Mean curvature flow · Shared terms “hyperbolic space”

  7. Topological Uniqueness for Self-expanders of Small Entropy

    Jacob Bernstein and Lu WangWider flows

    Topics Mean curvature flow · Same author Jacob Bernstein · Shared terms “Euclidean space”

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

    Jacob Bernstein and Lu Wang

    Topics Mean curvature flow · Same author Jacob Bernstein

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)

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.