Paper · Nov 20, 2020

Deforming a convex hypersurface by anisotropic curvature flows

Hongjie Ju, Boya Li, Yannan Liu

math.DGWider flowsarXiv:2011.11408
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Abstract

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem.

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Submitted
Nov 20, 2020
Subjects
math.DG, math.AP
Journal
Adv. Nonlinear Stud. (2020)
DOI
10.1515/ans-2020-2108

Comments: 19 pages. arXiv admin note: text overlap with arXiv:2005.02376

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 2016, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A flow method for a generalization of Christofell-Minkowski problem

    Boya Li, Hongjie Ju and Yannan LiuWider flows

    Topics Gauss curvature flow, Anisotropic & nonlocal flows, Long-time existence & convergence · Same authors Hongjie Ju, Boya Li, Yannan Liu · Shared terms “Christoffel Minkowski”, “smooth solution”

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

    YanNan Liu and Jian LuWider flows

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

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

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    Topics Gauss curvature flow, Other curvature flows · Shared terms “Christoffel Minkowski”, “elementary symmetric”, “convex hypersurfaces”

  4. A class of anisotropic expanding curvature flows

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

  5. 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 “Orlicz Christoffel”, “Christoffel Minkowski”

  6. Asymptotic convergence for a class of anisotropic curvature flows

    Haizhong Li, Botong Xu and Ruijia ZhangWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows · Shared terms “elementary symmetric”, “principal curvatures”

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

    Shanwei Ding and Guanghan LiWider flows

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

  8. 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 “Orlicz”, “Minkowski”

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)