Paper · Feb 21, 2022

Uniqueness of solutions to the logarithmic Minkowski problem in

Shibing Chen, Yibin Feng, Weiru Liu

math.APWider flowsarXiv:2202.10074
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Abstract

In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in without symmetry condition, provided the density of the measure is close to in norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in when the speed function is close to a positive constant.

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Submitted
Feb 21, 2022
Subjects
math.AP
Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsStability & uniqueness

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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 Uniqueness of closed self-similar solutions to the Gauss curvature flow (Kyeongsu Choi and Panagiota Daskalopoulos, 2016)8Choi 2016 Flow by Gauss curvature to Dual Orlicz-Minkowski problems (Li Chen, Qiang Tu, Di Wu et al., 2020)4Chen 2020 A flow method for a generalization of Lₚ Christofell-Minkowski problem (Boya Li, Hongjie Ju and Yannan Liu, 2022)6Li 2022 A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem (Shanwei Ding and Guanghan Li, 2022)7Ding 2022 Curvature bound for Lₚ Minkowski problem (Kyeongsu Choi, Minhyun Kim and Taehun Lee, 2023)1Choi 2023 Anisotropic Gauss curvature flow of complete non-compact graphs (Shujing Pan and Yong Wei, 2024)3Pan 2024 Flow by Gauss curvature to the Minkowski problem of p-harmonic measure (Chao Li and Xia Zhao, 2024)5Li 2024 The Lₚ chord Minkowski problem for super-critical exponent (Shibing Chen, Qi-Rui Li and Yuanyuan Li, 2025)2Chen 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows

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

  2. The chord Minkowski problem for super-critical exponent

    Shibing Chen, Qi-Rui Li and Yuanyuan LiWider flows

    Topics Gauss curvature flow, Other curvature flows · Same author Shibing Chen · Shared terms “Minkowski”

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

    Shujing Pan and Yong WeiWider flows

    Topics Gauss curvature flow, Anisotropic & nonlocal flows, Other curvature flows

  4. Flow by Gauss curvature to Dual Orlicz-Minkowski problems

    Li Chen, Qiang Tu, Di Wu et al.Wider flows

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

  5. Flow by Gauss curvature to the Minkowski problem of p-harmonic measure

    Chao Li and Xia ZhaoWider flows

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

  6. 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, Other curvature flows · Shared terms “Minkowski”

  7. 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, Other curvature flows · Shared terms “Minkowski”

  8. Uniqueness of closed self-similar solutions to the Gauss curvature flow

    Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)