Paper · Apr 30, 2020

Orlicz-Minkowski flows

Paul Bryan, Mohammad N. Ivaki, Julian Scheuer

math.DGWider flowsarXiv:2005.00143
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Abstract

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding version is the even -Minkowski problem for . Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the versions are the even -Minkowski problem for and the -Minkowski problem for . In the final section, we use a curvature flow with no global term to solve a class of -Christoffel-Minkowski type problems.

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Submitted
Apr 30, 2020
Subjects
math.DG, math.AP, math.MG
Journal
Calc. Var. Partial Differ. Equ. 60, no.1, (2021) art. 41
DOI
10.1007/s00526-020-01886-3

Comments: 30 pages

Other curvature flowsGauss curvature flowHomogeneous spacesLong-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 A unified flow approach to smooth, even Lₚ-Minkowski problems (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)2Bryan 2016 Flow by Gauss curvature to Dual Orlicz-Minkowski problems (Li Chen, Qiang Tu, Di Wu et al., 2020)8Chen 2020 Parabolic approaches to curvature equations (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 Christoffel-Minkowski flows (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)1Bryan 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)6Ding 2022 Capillary Orlicz-Minkowski flow in the upper half-space (Guanghan Li and Chenyang Liu, 2026)3Li 2026
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  1. 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 flow”, “Christoffel Minkowski”, “global term”

  2. A unified flow approach to smooth, even -Minkowski problems

    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”

  3. 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”, “Minkowski flow”, “stationary solutions”

  4. 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 “Orlicz Minkowski”

  5. 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 “Minkowski”, “global term”, “Christoffel”

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

    Shanwei Ding and Guanghan LiWider flows

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

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

    Shanwei Ding and Guanghan LiWider flows

    Topics Gauss curvature flow, Homogeneous spaces, Long-time existence & convergence · Shared terms “Orlicz”, “Christoffel Minkowski”, “global term”

  8. 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”, “anisotropic”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.