Paper · Jan 3, 2023

Non-normalized solutions to the horospherical Minkowski problem

Li Chen

math.APWider flowsarXiv:2301.01128v2
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Abstract

Recently, the horospherical -Minkowski problem in hyperbolic space was proposed as a counterpart of Minkowski problem in Euclidean space. Through designing a new volume preserving curvature flow, the existence of normalized even solution to the horospherical -Minkowski problem was solved for all . However, due to the lack of homogeneity of the horospherical -surface area measure, it is difficult to remove the normalizing factor. In this paper, we overcome this difficulty for by the degree-theoretic approach.

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Submitted
Jan 3, 2023
Last revised
Feb 20, 2023 (v2)
Subjects
math.AP

Comments: 12 pages

Other curvature flowsGauss curvature flowVolume-preserving & constrained flows

Its neighbourhood

Connected papers

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 Inverse Gauss curvature flows with free boundaries in a cone (Li Chen and Ni Xiang, 2018)7Chen 2018 A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1 (Li Chen, Qiang Tu and Ni Xiang, 2019)1Chen 2019 Flow by Gauss curvature to Dual Orlicz-Minkowski problems (Li Chen, Qiang Tu, Di Wu et al., 2020)3Chen 2020 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)2Wei 2022 Flow by Gauss curvature to the Minkowski problem of p-harmonic measure (Chao Li and Xia Zhao, 2024)5Li 2024 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)6Wei 2024 Gauss curvature flow to the Lₚ-Gaussian chord Minkowski problem (Xia Zhao and Peibiao Zhao, 2024)8Zhao 2024 Curvature measures and volume preserving flows in h-convex geometry (Xinxiu Zhang, 2026)4Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1

    Li Chen, Qiang Tu and Ni XiangWider flows

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

  2. Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Gauss curvature flow, Volume-preserving & constrained flows, Other curvature flows · Shared terms “hyperbolic space”

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

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

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

  4. Curvature measures and volume preserving flows in h-convex geometry

    Xinxiu ZhangWider flows

    Topics Gauss curvature flow, Volume-preserving & constrained flows, Other curvature flows · Shared terms “hyperbolic space”

  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. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Gauss curvature flow, Volume-preserving & constrained flows, Other curvature flows · Shared terms “hyperbolic space”

  7. Inverse Gauss curvature flows with free boundaries in a cone

    Li Chen and Ni XiangWider flows

    Topics Gauss curvature flow, Other curvature flows · Same author Li Chen

  8. Gauss curvature flow to the -Gaussian chord Minkowski problem

    Xia Zhao and Peibiao ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows · Shared terms “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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986)