Paper · May 4, 2020

A generalized Gauss curvature flow related to the Orlicz-Minkowski problem

YanNan Liu, Jian Lu

math.APWider flowsarXiv:2005.02376
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Abstract

In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean -space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a byproduct, two existence results of solutions to the even Orlicz-Minkowski problem are obtained, one of which improves the known result.

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Submitted
May 4, 2020
Subjects
math.AP, math.DG

Comments: 19 pages. arXiv admin note: text overlap with arXiv:2001.08862

Other curvature flowsGauss curvature flowLong-time existence & convergence

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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 Flow by Gauss curvature to Dual Orlicz-Minkowski problems (Li Chen, Qiang Tu, Di Wu et al., 2020)5Chen 2020 Orlicz-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)2Bryan 2020 Deforming a convex hypersurface by anisotropic curvature flows (Hongjie Ju, Boya Li and Yannan Liu, 2020)6Ju 2020 A Gauss curvature flow related to the Orlicz-Aleksandrov problem (Bin Chen and Peibiao Zhao, 2021)1Chen 2021 The chord Gauss curvature flow and its Lₚ chord Minkowski problem (Jinrong Hu, Yong Huang, Jian Lu et al., 2023)3Hu 2023 An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem (Bin Chen, Weidong Wang, Xia Zhao et al., 2023)4Chen 2023 Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity (Xia Zhao and Peibiao Zhao, 2024)7Zhao 2024 Capillary Orlicz-Minkowski flow in the upper half-space (Guanghan Li and Chenyang Liu, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Gauss curvature flow related to the Orlicz-Aleksandrov problem

    Bin Chen and Peibiao ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “flow related”, “Orlicz”

  2. Orlicz-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “Orlicz Minkowski”

  3. The chord Gauss curvature flow and its chord Minkowski problem

    Jinrong Hu, Yong Huang, Jian Lu et al.Wider flows

    Topics Gauss curvature flow, Other curvature flows · Same author Jian Lu · Shared terms “Minkowski”

  4. An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem

    Bin Chen, Weidong Wang, Xia Zhao et al.Wider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “Orlicz Minkowski”

  5. Flow by Gauss curvature to Dual Orlicz-Minkowski problems

    Li Chen, Qiang Tu, Di Wu et al.Wider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “Orlicz Minkowski”, “Euclidean”

  6. Deforming a convex hypersurface by anisotropic curvature flows

    Hongjie Ju, Boya Li and Yannan LiuWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same author YanNan Liu · Shared terms “Orlicz”, “convex hypersurfaces”, “support function”

  7. Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Orlicz Minkowski”

  8. Capillary Orlicz-Minkowski flow in the upper half-space

    Guanghan Li and Chenyang LiuWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “Orlicz Minkowski”

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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) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… 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)