Paper · Oct 3, 2017

Hyperbolic inverse mean curvature flow

Zhe Zhou, Chuan-Xi Wu, Jing Mao

math.DGWider flowsarXiv:1710.01242v2
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Abstract

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of () is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces have been shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic inverse mean curvature flow in the plane , whose evolving curves move normally.

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Submitted
Oct 3, 2017
Last revised
Nov 12, 2017 (v2)
Subjects
math.DG
Journal
Czech. Math. J. 70 (2020), 33-66
DOI
10.21136/CMJ.2019.0162-18

Comments: 35 pages. Hyperbolic inverse curvature flows in warped product manifolds have been discussed in [J. Mao, C.-X. Wu and Z. Zhou, Hyperbolic inverse curvature flows in warped products, preprint]. HIMCF of smooth strictly convex closed plane curves has been disscussed in v2 and some related interesting convergence and exanpansion results of the flow have also been shown. Comments are welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowHarmonic maps & DeTurck trick

Its neighbourhood

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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 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 Inverse mean curvature flow in complex hyperbolic space (Giuseppe Pipoli, 2016)3Pipoli 2016 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)8Hu 2016 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)2Chen 2017 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)5Chang 2018 A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone (Jing Mao and Qiang Tu, 2021)1Mao 2021 Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)6Gao 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 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)4Gao 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone

    Jing Mao and Qiang TuWider flows

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

  2. 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, Other curvature flows · Same author Jing Mao · Shared terms “evolving hypersurfaces”, “star shaped”

  3. Inverse mean curvature flow in complex hyperbolic space

    Giuseppe PipoliWider flows

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

  4. 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, Other curvature flows · Same author Jing Mao · Shared terms “evolving hypersurfaces”, “star shaped”, “mean convex”

  5. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  6. 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, Other curvature flows · Same author Jing Mao

  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, Other curvature flows · Same author Jing Mao

  8. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

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

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.