Paper · Apr 19, 2020

Shifted inverse curvature flows in hyperbolic space

Xianfeng Wang, Yong Wei, Tailong Zhou

math.DGWider flowsarXiv:2004.08822v2
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Abstract

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by with positive power for a smooth, symmetric, strictly increasing and -homogeneous curvature function of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.

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Submitted
Apr 19, 2020
Last revised
Feb 2, 2023 (v2)
Subjects
math.DG, math.AP
Journal
Calculus of Variations and Partial Differential Equations, vol.62, article no. 93 (2023)
DOI
10.1007/s00526-023-02429-2

Comments: 41 pages, 2 figures

Other curvature flowsInverse mean curvature flowHomogeneous spaces

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Yingxiang Hu, Haizhong Li, Yong Wei et al.Wider flows

    Topics Other curvature flows · Same authors Yong Wei, Tailong Zhou · Shared terms “hyperbolic space”, “positive power”, “approached”

  2. New pinching estimates for Inverse curvature flows in space forms

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    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “principal curvatures”, “convex hypersurfaces”

  3. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Homogeneous spaces, Other curvature flows · Same authors Yong Wei, Tailong Zhou · Shared terms “hyperbolic space”, “convex hypersurfaces”

  4. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same authors Yong Wei, Tailong Zhou · Shared terms “Wang”

  5. Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Other curvature flows · Same authors Yong Wei, Tailong Zhou · Shared terms “hyperbolic space”, “positive power”, “convex hypersurfaces”

  6. Quermassintegral preserving curvature flow in Hyperbolic space

    Ben Andrews and Yong WeiWider flows

    Topics Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “strictly increasing”, “positive power”

  7. Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space

    Ben Andrews, Xuzhong Chen and Yong WeiWider flows

    Topics Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “positive power”, “shifted principal”

  8. Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

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    Topics Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic space”, “principal curvatures”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.