Paper · Aug 15, 2024

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

Ya Gao, Jing Mao

math.DGWider flowsarXiv:2408.07949
PDF Abstract page

As on arXiv

Abstract

For a given smooth convex cone in the Euclidean -space which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to , where is a positive function of the radial distance parameter and is the mean curvature of the evolving hypersurfaces. The evolution of those hypersurfaces inside the cone yields a fully nonlinear parabolic Neumann problem. Under suitable constraints on the first and the second derivatives of the radial function , we can prove the long-time existence of this flow, and moreover the evolving hypersurfaces converge smoothly to a piece of the round sphere.

Comments: 17 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.08884, arXiv:2104.10600, arXiv:2106.05973

Other curvature flowsInverse mean curvature flowFree boundary & capillaryLong-time existence & convergence

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 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 Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)2Gao 2021 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 On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space (Fang Hong, 2021)6Hong 2021 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)3Chai 2022 A class of generalized fully nonlinear curvature flows and its applications (Jinrong Hu, Jiaqian Liu, Di Ma et al., 2022)7Hu 2022 Two nonlocal inverse curvature flows of convex closed plane curves (Zezhen Sun, 2023)8Sun 2023 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)5Pan 2025 Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows (Weimin Sheng and Jiazhuo Yang, 2025)4Sheng 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  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, Inverse mean curvature flow, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “Neumann”, “converges smoothly”

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

    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”

  3. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows PDF

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “hypersurface inside”, “convex hypersurfaces”

  4. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “star shaped”, “converges smoothly”

  5. Inverse curvature flows for capillary hypersurfaces in the unit ball

    Shujing Pan and Bo YangWider flows PDF

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “convex hypersurfaces”, “Euclidean”

  6. On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

    Fang HongWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “converges smoothly”, “mean convex”

  7. A class of generalized fully nonlinear curvature flows and its applications

    Jinrong Hu, Jiaqian Liu, Di Ma et al.Wider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “positive function”, “constraints”

  8. Two nonlocal inverse curvature flows of convex closed plane curves

    Zezhen SunWider flows PDF

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “converges smoothly”

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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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)