Paper · Oct 6, 2016

Inverse mean curvature flow in complex hyperbolic space

Giuseppe Pipoli

math.DGWider flowsarXiv:1610.01886v4
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Abstract

We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.

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Submitted
Oct 6, 2016
Last revised
Mar 27, 2018 (v4)
Subjects
math.DG

Comments: 31 pages, minor changes. This is the final version to appear on Annales scientifiques de l'ENS

Mean curvature flowOther curvature flowsInverse mean curvature flow

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 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)8Mullins 2016 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)3Hu 2016 Inverse mean curvature flow in quaternionic hyperbolic space (Giuseppe Pipoli, 2017)1Pipoli 2017 Hyperbolic inverse mean curvature flow (Zhe Zhou, Chuan-Xi Wu and Jing Mao, 2017)5Zhou 2017 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)6Chang 2018 Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces (Brian Harvie, 2019)7Harvie 2019 Nonhomogeneous expanding flows in hyperbolic spaces (Giuseppe Pipoli, 2020)2Pipoli 2020 A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone (Jing Mao and Qiang Tu, 2021)4Mao 2021
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 in quaternionic hyperbolic space

    Giuseppe PipoliWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Giuseppe Pipoli · Shared terms “hyperbolic space”, “star shaped”, “hypersurface stays”

  2. Nonhomogeneous expanding flows in hyperbolic spaces

    Giuseppe PipoliWider flows

    Topics Other curvature flows · Same author Giuseppe Pipoli · Shared terms “hyperbolic space”, “conformal multiple”, “standard sub”

  3. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

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

  4. 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 · Shared terms “star shaped”, “shaped hypersurfaces”, “evolving hypersurfaces”

  5. Hyperbolic inverse mean curvature flow

    Zhe Zhou, Chuan-Xi Wu and Jing MaoWider flows

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

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

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  7. Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

    Brian HarvieWider flows

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

  8. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

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

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)

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.