Paper · Jun 27, 2022

Relative Entropy of Hypersurfaces in Hyperbolic Space

Junfu Yao

math.DGWider flowsarXiv:2206.13141
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Abstract

We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham-Witten[RW99]. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space.

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Submitted
Jun 27, 2022
Subjects
math.DG
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 2021, 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 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)1Huang 2021 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)4Chai 2022 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)8White 2022 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)6Laux 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)5Laux 2022 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)7Mei 2024 Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces (Xiaobo Liu and Wanxu Yang, 2025)2Liu 2025 Contraction of hypersurfaces with positive sectional curvature in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)3Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Mean curvature flow · Shared terms “hyperbolic space”

  2. Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces

    Xiaobo Liu and Wanxu YangWider flows

    Topics Mean curvature flow · Shared terms “hyperbolic space”

  3. Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

    Tianci Luo, Yong Wei and Rong ZhouWider flows

    Topics Mean curvature flow · Shared terms “hyperbolic space”

  4. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows

    Topics Mean curvature flow · Shared terms “hyperbolic space”

  5. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows

    Topics Mean curvature flow · Shared terms “relative entropy”

  6. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows

    Topics Mean curvature flow · Shared terms “relative entropy”

  7. Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

    Xinqun Mei and Liangjun WengWider flows

    Shared terms “hyperbolic space”

  8. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows

    Topics Mean curvature flow · Shared terms “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. 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.