Paper · Feb 15, 2018

Inverse Gauss curvature flows with free boundaries in a cone

Li Chen, Ni Xiang

math.DGWider flowsarXiv:1802.05413v2
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Abstract

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere after rescaling.

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Submitted
Feb 15, 2018
Last revised
Aug 7, 2020 (v2)
Subjects
math.DG, math.AP

Comments: 31 pages

Other curvature flowsGauss curvature flowFree boundary & capillaryLong-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 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)1Chen 2017 A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1 (Li Chen, Qiang Tu and Ni Xiang, 2019)3Chen 2019 Flow by Gauss curvature to Dual Orlicz-Minkowski problems (Li Chen, Qiang Tu, Di Wu et al., 2020)2Chen 2020 Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)4Gao 2021 Gauss curvature flow with shrinking obstacle (Ki-Ahm Lee and Taehun Lee, 2023)6Lee 2023 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)8Gao 2024 Smoothness up to the free boundary for the p-Laplacian evolution equation and the α-Gauss curvature flow (Albert Chau and Ben Weinkove, 2024)7Chau 2024 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)5Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse mean curvature flow inside a cone in warped products

    Li Chen, Jing Mao, Ni Xiang et al.Wider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Li Chen, Ni Xiang · Shared terms “cone perpendicularly”, “evolution exists”, “convex cone”

  2. 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 · Same authors Li Chen, Ni Xiang · Shared terms “strictly convex”, “converges smoothly”

  3. 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 authors Li Chen, Ni Xiang · Shared terms “strictly convex”, “convex hypersurfaces”

  4. Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Gauss curvature flow, Free boundary & capillary, Long-time existence & convergence · Shared terms “inverse Gauss”, “hypersurfaces converge”, “converges smoothly”

  5. Free boundary flows by powers of the Gauss curvature in the unit ball

    Tianci Luo, Yong Wei and Rong ZhouWider flows

    Topics Gauss curvature flow, Free boundary & capillary, Long-time existence & convergence · Shared terms “strictly convex”, “hypersurfaces converge”, “converges smoothly”

  6. Gauss curvature flow with shrinking obstacle

    Ki-Ahm Lee and Taehun LeeWider flows

    Topics Gauss curvature flow, Free boundary & capillary, Long-time existence & convergence

  7. Smoothness up to the free boundary for the -Laplacian evolution equation and the -Gauss curvature flow

    Albert Chau and Ben WeinkoveWider flows

    Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows

  8. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Other curvature flows · Shared terms “evolving hypersurfaces”, “cone perpendicularly”, “convex cone”

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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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) 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)