Paper · Apr 18, 2017

Inverse mean curvature flow in quaternionic hyperbolic space

Giuseppe Pipoli

math.DGWider flowsarXiv:1704.05227v2
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Abstract

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic 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 defined on a codimension 3 distribution. Finally we show that there exists a family of examples such that the qc-scalar curvature of this sub-Riemannian limit is not constant.

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Submitted
Apr 18, 2017
Last revised
Oct 19, 2017 (v2)
Subjects
math.DG

Comments: 20 pages. arXiv admin note: substantial text overlap with arXiv:1610.01886

Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvatureBoundaries & non-compact

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 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)1Pipoli 2016 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)3Hu 2016 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 Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3 (Yuguang Shi and Jintian Zhu, 2018)7Shi 2018 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 A Minkowski type inequality in warped cylinders (Shujing Pan and Bo Yang, 2024)8Pan 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 in complex 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 Scalar curvature, Other curvature flows · Same author Giuseppe Pipoli · Shared terms “quaternionic hyperbolic”, “hyperbolic space”, “conformal multiple”

  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 “shaped hypersurfaces”, “star shaped”, “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. Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension

    Yuguang Shi and Jintian ZhuWider flows

    Topics Inverse mean curvature flow, 3-manifolds, Boundaries & non-compact · Shared terms “star shaped”

  8. A Minkowski type inequality in warped cylinders

    Shujing Pan and Bo YangWider flows

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Shared terms “shaped hypersurfaces”, “star shaped”, “hyperbolic space”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.