Paper · Aug 9, 2016

A unified flow approach to smooth, even -Minkowski problems

Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

math.DGWider flowsarXiv:1608.02770v2
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Abstract

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even -Minkowski problems in for

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Submitted
Aug 9, 2016
Last revised
May 6, 2018 (v2)
Subjects
math.DG, math.AP
Journal
Analysis & PDE 12 (2019) 259-280
DOI
10.2140/apde.2019.12.259

Comments: 21 pages. Comments are welcome

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 A class of anisotropic expanding curvature flows (Weimin Sheng and Caihong Yi, 2019)5Sheng 2019 Parabolic approaches to curvature equations (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)3Bryan 2020 Orlicz-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)1Bryan 2020 A generalized Gauss curvature flow related to the Orlicz-Minkowski problem (YanNan Liu and Jian Lu, 2020)7Liu 2020 Christoffel-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)2Bryan 2020 An expanding curvature flow and the (p,q)-Christoffel-Minkowski problems (Bin Chen, Jingshi Cui and Peibiao Zhao, 2022)8Chen 2022 A flow method for a generalization of Lₚ Christofell-Minkowski problem (Boya Li, Hongjie Ju and Yannan Liu, 2022)4Li 2022 Uniqueness of solutions to a class of isotropic curvature problems (Mohammad N. Ivaki and Emanuel Milman, 2023)6Ivaki 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Orlicz-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Minkowski”, “stationary”, “anisotropic”

  2. Christoffel-Minkowski flows

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “Minkowski”

  3. Parabolic approaches to curvature equations

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Gauss curvature flow, Other curvature flows · Same authors Paul Bryan, Mohammad N. Ivaki, Julian Scheuer · Shared terms “prescribed curvature”, “Minkowski”, “Guan”

  4. A flow method for a generalization of Christofell-Minkowski problem

    Boya Li, Hongjie Ju and Yannan LiuWider flows

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

  5. A class of anisotropic expanding curvature flows

    Weimin Sheng and Caihong YiWider flows

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

  6. Uniqueness of solutions to a class of isotropic curvature problems

    Mohammad N. Ivaki and Emanuel MilmanWider flows

    Topics Gauss curvature flow, Other curvature flows · Same author Mohammad N. Ivaki · Shared terms “Minkowski”

  7. 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 · Shared terms “Minkowski”

  8. 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 “expanding curvature”, “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.

Foundational paper in the Gauss curvature flow guide: Bryan–Ivaki–Scheuer: one flow approach to the smooth, even -Minkowski problems.

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)