Paper · May 9, 2021

Geometric inequalities for static convex domains in hyperbolic space

Yingxiang Hu, Haizhong Li

math.DGWider flowsarXiv:2105.03911
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.

View PDF View HTML arXiv page
Submitted
May 9, 2021
Subjects
math.DG

Comments: 25 pages, no figures. All comments are welcome

Other curvature flowsVolume-preserving & constrained flowsGeometric 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 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)5Hu 2016 Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space (Yingxiang Hu and Haizhong Li, 2018)3Hu 2018 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)4Andrews 2019 Contraction of surfaces in hyperbolic space and in sphere (Yingxiang Hu, Haizhong Li, Yong Wei et al., 2019)8Hu 2019 Locally constrained curvature flows and geometric inequalities in hyperbolic space (Yingxiang Hu, Haizhong Li and Yong Wei, 2020)1Hu 2020 The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces (Shujing Pan and Bo Yang, 2023)2Pan 2023 New quermassintegral and Poincaré type inequalities for non-convex domains (Yingxiang Hu and Mohammad N. Ivaki, 2024)6Hu 2024 New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms (Jie Wu, 2025)7Wu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Locally constrained curvature flows and geometric inequalities in hyperbolic space

    Yingxiang Hu, Haizhong Li and Yong WeiWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Other curvature flows · Same authors Yingxiang Hu, Haizhong Li · Shared terms “hyperbolic space”, “locally constrained”, “preserved along”

  2. The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

    Shujing Pan and Bo YangWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Other curvature flows · Shared terms “static convex”, “convex domains”, “hyperbolic space”

  3. Geometric inequalities for hypersurfaces with nonnegative sectional curvature 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”

  4. Harmonic mean curvature flow and geometric inequalities

    Ben Andrews, Yingxiang Hu and Haizhong LiWider flows

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

  5. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

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

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

    Yingxiang Hu and Mohammad N. IvakiWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Other curvature flows · Same author Yingxiang Hu · Shared terms “convex domains”, “locally constrained”

  7. New weighted Alexandrov-Fenchel type inequalities and Minkowski inequalities in space forms

    Jie WuWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows, Other curvature flows · Shared terms “static convex”, “hyperbolic space”, “locally constrained”

  8. Contraction of surfaces in hyperbolic space and in sphere

    Yingxiang Hu, Haizhong Li, Yong Wei et al.Wider flows

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

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.

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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)