Paper · Oct 4, 2023

Gauss curvature flow with shrinking obstacle

Ki-Ahm Lee, Taehun Lee

math.DGWider flowsarXiv:2310.02668
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Abstract

We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the regularity of free boundaries under a uniform thickness assumption.

Comments: To appear in Math. Ann

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 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 Curvature bound for Lₚ Minkowski problem (Kyeongsu Choi, Minhyun Kim and Taehun Lee, 2023)2Choi 2023 Long time regularity of the p-Gauss curvature flow with flat side (G. Huang, X. -J. Wang and Y. Zhou, 2024)6Huang 2024 Anisotropic Gauss curvature flow of complete non-compact graphs (Shujing Pan and Yong Wei, 2024)8Pan 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)4Huang 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)3Chau 2024 The capillary Gauss curvature flow (Xinqun Mei, Guofang Wang and Liangjun Weng, 2025)5Mei 2025 Capillary Orlicz-Minkowski flow in the upper half-space (Guanghan Li and Chenyang Liu, 2026)7Li 2026 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)1Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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

  2. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Taehun Lee · Shared terms “curvature bounds”

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

    Albert Chau and Ben WeinkoveWider flows PDF

    Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows · Shared terms “time existence”

  4. Regularity of the -Gauss curvature flow with flat side

    Genggeng Huang, Xu-Jia Wang and Yang ZhouWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  5. The capillary Gauss curvature flow

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows PDF

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

  6. Long time regularity of the -Gauss curvature flow with flat side

    G. Huang, X. -J. Wang and Y. ZhouWider flows PDF

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

  7. Capillary Orlicz-Minkowski flow in the upper half-space

    Guanghan Li and Chenyang LiuWider flows PDF

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

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

    Shujing Pan and Yong WeiWider flows PDF

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

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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)