Paper · Apr 21, 2022

A flow method for a generalization of Christofell-Minkowski problem

Boya Li, Hongjie Ju, Yannan Liu

math.DGWider flowsarXiv:2204.09933
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Abstract

In this paper, a generalization of the -Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for .

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Submitted
Apr 21, 2022
Subjects
math.DG, math.AP
Journal
Communications on Pure and Applied Analysis, 21 (3) 2022, 785-796
DOI
10.3934/cpaa.2021198

Comments: arXiv admin note: substantial text overlap with arXiv:2011.11408

Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsLong-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 2021, 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 A class of inverse curvature flows and L^p dual Christoffel-Minkowski problem (Shanwei Ding and Guanghan Li, 2022)2Ding 2022 An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems (Bin Chen, Jingshi Cui and Peibiao Zhao, 2022)1Chen 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)5Ding 2022 Flow by Gauss curvature to the Lₚ-Gaussian Minkowski problem (Weimin Sheng and Ke Xue, 2022)7Sheng 2022 Curvature bound for Lₚ Minkowski problem (Kyeongsu Choi, Minhyun Kim and Taehun Lee, 2023)4Choi 2023 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)8Bellettini 2023 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)6Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems

    Bin Chen, Jingshi Cui and Peibiao ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”, “generalization”, “smooth solutions”

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

    Shanwei Ding and Guanghan LiWider flows

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

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

    Shanwei Ding and Guanghan LiWider flows

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

  4. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows

    Topics Anisotropic & nonlocal flows, Gauss curvature flow, Other curvature flows · Shared terms “Minkowski”

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

    Shanwei Ding and Guanghan LiWider flows

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

  6. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows

  7. Flow by Gauss curvature to the -Gaussian Minkowski problem

    Weimin Sheng and Ke XueWider flows

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

  8. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

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) 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)