Paper · Apr 10, 2021

A Class Of Curvature Flows Expanded By Support Function And Curvature Function In The Euclidean Space And Hyperbolic Space

Shanwei Ding, Guanghan Li

math.DGWider flowsarXiv:2104.04783
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Abstract

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space with speed , where is the support function of the hypersurface, is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For , we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt and Urbas can be recovered by putting and in our first result. If the initial hypersurface is convex, this is our previous work. If and the ambient space is hyperbolic space , we prove that the flow has a longtime existence and smooth convergence to a coordinate slice. The flow in is equivalent (up to an isomorphism) to a re-parametrization of the original flow in case. Finally, we find a family of monotone quantities along the flows in . As applications, we give a new proof of a family of inequalities involving the weighted integral of th elementary symmetric function for -convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in.

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Submitted
Apr 10, 2021
Subjects
math.DG
Other curvature flowsLong-time existence & convergenceGeometric inequalities

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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)7Hu 2016 Quermassintegral preserving curvature flow in Hyperbolic space (Ben Andrews and Yong Wei, 2017)4Andrews 2017 A class of curvature flows expanded by support function and curvature function (Shanwei Ding and Guanghan Li, 2020)1Ding 2020 A class of inverse curvature flows and L^p dual Christoffel-Minkowski problem (Shanwei Ding and Guanghan Li, 2022)2Ding 2022 Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem (Shanwei Ding and Guanghan Li, 2022)3Ding 2022 A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem (Shanwei Ding and Guanghan Li, 2022)6Ding 2022 Inverse Hessian Curvature Flow in Minkowski Space I: Cocompact Hypersurfaces (Dake Li, Zhizhang Wang and Shiqi Yin, 2026)8Li 2026 The quermassintegral inequalities for k-convex spiral-star-shaped domains (Junfang Li, 2026)5Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A class of curvature flows expanded by support function and curvature function

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li · Shared terms “flow expanded”, “support function”, “curvature function”

  2. A class of inverse curvature flows and dual Christoffel-Minkowski problem

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li · Shared terms “support function”, “shaped hypersurfaces”, “positive function”

  3. Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

    Shanwei Ding and Guanghan LiWider flows

    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li · Shared terms “inequalities involving”, “quermassintegral”, “equivalent”

  4. Quermassintegral preserving curvature flow in Hyperbolic space

    Ben Andrews and Yong WeiWider flows

    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Shared terms “hyperbolic space”, “quermassintegral”, “initial hypersurface”

  5. The quermassintegral inequalities for k-convex spiral-star-shaped domains

    Junfang LiWider flows

    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Shared terms “star shaped”, “quermassintegral inequalities”, “support function”

  6. A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

    Shanwei Ding and Guanghan LiWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Shanwei Ding, Guanghan Li · Shared terms “extension”

  7. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

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

  8. Inverse Hessian Curvature Flow in Minkowski Space I: Cocompact Hypersurfaces

    Dake Li, Zhizhang Wang and Shiqi YinWider flows

    Topics Geometric inequalities, Long-time existence & convergence, Other curvature flows · Shared terms “quermassintegral inequalities”, “longtime existence”

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)