Paper · Jan 21, 2020

Flow by Gauss curvature to Dual Orlicz-Minkowski problems

Li Chen, Qiang Tu, Di Wu, Ni Xiang

math.APWider flowsarXiv:2001.07324
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Abstract

In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument provides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.

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Submitted
Jan 21, 2020
Subjects
math.AP

Comments: 14pages

Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsLong-time existence & convergence

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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 Inverse Gauss curvature flows with free boundaries in a cone (Li Chen and Ni Xiang, 2018)2Chen 2018 A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1 (Li Chen, Qiang Tu and Ni Xiang, 2019)3Chen 2019 Orlicz-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)5Bryan 2020 A generalized Gauss curvature flow related to the Orlicz-Minkowski problem (YanNan Liu and Jian Lu, 2020)4Liu 2020 Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem (Shanwei Ding and Guanghan Li, 2022)7Ding 2022 A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem (Shanwei Ding and Guanghan Li, 2022)1Ding 2022 An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem (Bin Chen, Weidong Wang, Xia Zhao et al., 2023)6Chen 2023 Anisotropic Gauss curvature flow of complete non-compact graphs (Shujing Pan and Yong Wei, 2024)8Pan 2024
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  1. A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem

    Shanwei Ding and Guanghan LiWider flows

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

  2. Inverse Gauss curvature flows with free boundaries in a cone

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  3. A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1

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    Topics Gauss curvature flow, Other curvature flows · Same authors Li Chen, Qiang Tu, Ni Xiang · Shared terms “Minkowski”, “strictly convex”

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

  5. Orlicz-Minkowski flows

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

  6. 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”, “convex solution”, “type equations”

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

    Shanwei Ding and Guanghan LiWider flows

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

  8. Anisotropic Gauss curvature flow of complete non-compact graphs

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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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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)