Paper · Mar 14, 2019

Harmonic mean curvature flow and geometric inequalities

Ben Andrews, Yingxiang Hu, Haizhong Li

math.DGWider flowsarXiv:1903.05903
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Abstract

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in hyperbolic space, which was previously proved for horospherically convex hypersurfaces by Wang and Xia [44]. Finally, we use the mean curvature flow to prove a new Heintze-Karcher type inequality for hypersurfaces with positive Ricci curvature in hyperbolic space.

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Submitted
Mar 14, 2019
Subjects
math.DG, math.AP

Comments: All comments are welcome!

Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalities

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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 An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere (Frederico Girão and Neilha M. Pinheiro, 2016)3Girão 2016 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)4Hu 2016 A Penrose type inequaltiy for graphs over Reissner-Nordström-anti-deSitter manifold (Daguang Chen, Haizhong Li and Tailong Zhou, 2017)8Chen 2017 Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space (Ben Andrews, Xuzhong Chen and Yong Wei, 2018)5Andrews 2018 Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space (Yingxiang Hu and Haizhong Li, 2018)1Hu 2018 Locally constrained curvature flows and geometric inequalities in hyperbolic space (Yingxiang Hu, Haizhong Li and Yong Wei, 2020)2Hu 2020 Geometric inequalities for static convex domains in hyperbolic space (Yingxiang Hu and Haizhong Li, 2021)6Hu 2021 New quermassintegral and Poincaré type inequalities for non-convex domains (Yingxiang Hu and Mohammad N. Ivaki, 2024)7Hu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 authors Yingxiang Hu, Haizhong Li · Shared terms “hyperbolic space”, “Alexandrov Fenchel”, “type inequality”

  2. 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 authors Yingxiang Hu, Haizhong Li · Shared terms “hyperbolic space”, “convex hypersurfaces”, “Fenchel type”

  3. An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere

    Frederico Girão and Neilha M. PinheiroWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Fenchel type”, “Alexandrov Fenchel”, “type inequality”

  4. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

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

  5. Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Geometric inequalities, Other curvature flows · Same author Ben Andrews · Shared terms “horospherically convex”, “Fenchel type”, “Alexandrov Fenchel”

  6. Geometric inequalities for static convex domains in hyperbolic space

    Yingxiang Hu and Haizhong LiWider flows

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

  7. New quermassintegral and Poincaré type inequalities for non-convex domains

    Yingxiang Hu and Mohammad N. IvakiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Yingxiang Hu · Shared terms “type inequality”, “Fenchel type”, “Alexandrov Fenchel”

  8. A Penrose type inequaltiy for graphs over Reissner-Nordström-anti-deSitter manifold

    Daguang Chen, Haizhong Li and Tailong ZhouWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Same author Haizhong Li · Shared terms “Alexandrov Fenchel”, “type inequality”, “Fenchel inequalities”

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.