Paper · Apr 18, 2021

A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone

Jing Mao, Qiang Tu

math.DGWider flowsarXiv:2104.08884v2
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Abstract

Given a smooth convex cone in the Euclidean -space (), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a class of inverse curvature flows, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.

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Submitted
Apr 18, 2021
Last revised
Apr 20, 2021 (v2)
Subjects
math.DG

Comments: 17 pages. This paper was finished in January 2019 when the first author visited IST, University of Lisbon. We just put it on arXiv very recently (the status of two references has been updated). Several related works will also be put on arXiv soon. Comments are welcome. Two typos have been corrected in Theorem 1.1

Other curvature flowsInverse mean curvature flowLong-time existence & convergence

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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 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 Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold (Li Chen and Jing Mao, 2016)3Chen 2016 Inverse mean curvature flow in complex hyperbolic space (Giuseppe Pipoli, 2016)6Pipoli 2016 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)1Chen 2017 Hyperbolic inverse mean curvature flow (Zhe Zhou, Chuan-Xi Wu and Jing Mao, 2017)4Zhou 2017 Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)5Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)7Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ℝ²₁ (Ya Gao, Chenyang Liu and Jing Mao, 2021)8Gao 2021 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)2Gao 2024
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 Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “cone evolve”, “star shaped”, “cone perpendicularly”

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

    Ya Gao and Jing MaoWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “evolving hypersurfaces”, “star shaped”, “cone perpendicularly”

  3. Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold

    Li Chen and Jing MaoWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “star shaped”, “hypersurfaces converge”

  4. Hyperbolic inverse mean curvature flow

    Zhe Zhou, Chuan-Xi Wu and Jing MaoWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Jing Mao · Shared terms “star shaped”, “evolving hypersurfaces”, “mean convex”

  5. Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “hypersurfaces converge”, “time tends”, “converges smoothly”

  6. Inverse mean curvature flow in complex hyperbolic space

    Giuseppe PipoliWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “star shaped”, “shaped hypersurfaces”, “evolving hypersurfaces”

  7. An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “hypersurfaces converge”, “time tends”, “converges smoothly”

  8. An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

    Ya Gao, Chenyang Liu and Jing MaoWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “time tends”, “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) 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)