Paper · Aug 19, 2021

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

Ya Gao, Jing Mao

math.DGWider flowsarXiv:2108.08686
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Abstract

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane , of center at origin and radius , in the -dimensional Lorentz-Minkowski space along the inverse Gauss curvature flow (i.e., the evolving speed equals the -th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of as time tends to infinity.

Comments: This work has been announced in our previous work [arXiv:2104.10600v4]. 31 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2106.05973

Other curvature flowsGauss curvature flowBoundaries & non-compactFree boundary & capillaryLong-time existence & convergencePhysics

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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 2021, 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 An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ℝ²₁ (Ya Gao, Chenyang Liu and Jing Mao, 2021)1Gao 2021 Entire convex curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)8Wang 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)5Wang 2022 Flow by Gauss curvature to the Lₚ-Gaussian Minkowski problem (Weimin Sheng and Ke Xue, 2022)7Sheng 2022 An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem (Bin Chen, Weidong Wang, Xia Zhao et al., 2023)3Chen 2023 Gauss curvature flow with shrinking obstacle (Ki-Ahm Lee and Taehun Lee, 2023)4Lee 2023 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)2Gao 2024 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)6Luo 2026
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  1. An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

    Ya Gao, Chenyang Liu and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Physics, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “spacelike graphic”, “Lorentz Minkowski”, “evolving spacelike”

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

    Ya Gao and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Long-time existence & convergence, Other curvature flows · Same authors Ya Gao, Jing Mao · Shared terms “speed equal”, “hypersurfaces converge”, “Neumann”

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  4. Gauss curvature flow with shrinking obstacle

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

  5. Entire curvature flow in Minkowski space

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    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Minkowski space”, “spacelike”, “flow exists”

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

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

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

  7. Flow by Gauss curvature to the -Gaussian Minkowski problem

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

  8. Entire convex curvature flow in Minkowski space

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    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Minkowski space”, “spacelike”, “flow exists”

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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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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)