Paper · Oct 31, 2016

Willmore inequality on hypersurfaces in hyperbolic space

Yingxiang Hu

math.DGWider flowsarXiv:1610.09794
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Abstract

In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic -space.

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Submitted
Oct 31, 2016
Subjects
math.DG

Comments: All comments are welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalities

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 A geometric inequality for convex free boundary hypersurfaces in the unit ball (Ben Lambert and Julian Scheuer, 2016)7Lambert 2016 Inverse mean curvature flow in complex hyperbolic space (Giuseppe Pipoli, 2016)6Pipoli 2016 Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space (Yingxiang Hu and Haizhong Li, 2018)2Hu 2018 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)1Andrews 2019 Locally constrained curvature flows and geometric inequalities in hyperbolic space (Yingxiang Hu, Haizhong Li and Yong Wei, 2020)4Hu 2020 Geometric inequalities for static convex domains in hyperbolic space (Yingxiang Hu and Haizhong Li, 2021)3Hu 2021 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)5Chai 2022 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)8Pan 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Harmonic mean curvature flow and geometric inequalities

    Ben Andrews, Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Yingxiang Hu · Shared terms “hyperbolic space”, “convex hypersurfaces”

  2. Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

    Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Yingxiang Hu · Shared terms “hyperbolic space”

  3. Geometric inequalities for static convex domains in hyperbolic space

    Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Other curvature flows · Same author Yingxiang Hu · Shared terms “hyperbolic space”

  4. Locally constrained curvature flows and geometric inequalities in hyperbolic space

    Yingxiang Hu, Haizhong Li and Yong WeiWider flows

    Topics Geometric inequalities, Other curvature flows · Same author Yingxiang Hu · Shared terms “hyperbolic space”, “convex hypersurfaces”

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

    Xiaoxiang ChaiWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “hyperbolic space”, “Willmore”, “convex hypersurfaces”

  6. Inverse mean curvature flow in complex hyperbolic space

    Giuseppe PipoliWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “hyperbolic space”, “star shaped”, “mean convex”

  7. A geometric inequality for convex free boundary hypersurfaces in the unit ball

    Ben Lambert and Julian ScheuerWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Willmore”, “convex hypersurfaces”

  8. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “hyperbolic space”, “star shaped”, “mean convex”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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.