Paper · Apr 16, 2024

Anisotropic Gauss curvature flow of complete non-compact graphs

Shujing Pan, Yong Wei

math.DGWider flowsarXiv:2404.10298
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Abstract

In this paper, we consider the anisotropic -Gauss curvature flow for complete noncompact convex hypersurfaces in the Euclidean space with the anisotropy determined by a smooth closed uniformly convex Wulff shape. We show that for all positive power , if the initial hypersurface is complete noncompact and locally uniformly convex, then the solution of the flow exists for all positive time.

Comments: 23 pages. All comments are welcome

Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsBoundaries & non-compactLong-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 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 A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem (Shanwei Ding and Guanghan Li, 2022)5Ding 2022 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)1Wei 2022 Curvature bound for Lₚ Minkowski problem (Kyeongsu Choi, Minhyun Kim and Taehun Lee, 2023)2Choi 2023 Gauss curvature flow with shrinking obstacle (Ki-Ahm Lee and Taehun Lee, 2023)7Lee 2023 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)3Wei 2024 Asymptotic behaviour of the weak inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2025)6Gao 2025 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)4Luo 2026 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)8Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “convex hypersurfaces”, “positive power”, “positive time”

  2. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows PDF

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

  3. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “positive time”, “convex hypersurfaces”

  4. 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, Long-time existence & convergence, Other curvature flows · Same author Yong Wei · Shared terms “convex hypersurfaces”, “Euclidean”

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

    Shanwei Ding and Guanghan LiWider flows PDF

    Topics Anisotropic & nonlocal flows, Gauss curvature flow, Long-time existence & convergence

  6. Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows PDF

    Topics Anisotropic & nonlocal flows · Same author Yong Wei · Shared terms “Wulff shape”

  7. Gauss curvature flow with shrinking obstacle

    Ki-Ahm Lee and Taehun LeeWider flows PDF

    Topics Gauss curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “positive power”

  8. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence, Other curvature flows · Shared terms “Wulff shape”, “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) 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) 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)