Paper · May 13, 2017

Inverse mean curvature flow inside a cone in warped products

Li Chen, Jing Mao, Ni Xiang, Chi Xu

math.DGWider flowsarXiv:1705.04865v3
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Abstract

Given a convex cone in the prescribed warped product, we consider 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 along the inverse mean curvature flow, then, by using the convexity of the cone, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere as time tends to infinity.

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Submitted
May 13, 2017
Last revised
Jun 1, 2017 (v3)
Subjects
math.DG

Comments: 12 pages. Several minor revisions (including typos) have been made to v2

Mean curvature flowOther curvature flowsInverse mean curvature flowLong-time existence & convergenceRotational symmetry & warped products

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 flows in warped product manifolds (Hengyu Zhou, 2016)6Zhou 2016 Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold (Li Chen and Jing Mao, 2016)5Chen 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)4Mullins 2016 Hyperbolic inverse mean curvature flow (Zhe Zhou, Chuan-Xi Wu and Jing Mao, 2017)7Zhou 2017 Inverse Gauss curvature flows with free boundaries in a cone (Li Chen and Ni Xiang, 2018)2Chen 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)8Gao 2021 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)3Gao 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, Long-time existence & convergence, Other curvature flows · Same author Jing Mao · Shared terms “cone evolve”, “star shaped”, “cone perpendicularly”

  2. Inverse Gauss curvature flows with free boundaries in a cone

    Li Chen and Ni XiangWider flows

    Topics Long-time existence & convergence, Other curvature flows · Same authors Li Chen, Ni Xiang · Shared terms “cone perpendicularly”, “evolution exists”, “convex cone”

  3. 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”, “cone perpendicularly”, “hypersurfaces inside”

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

    Thomas MullinsWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “star shaped”

  5. 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 authors Li Chen, Jing Mao · Shared terms “hypersurfaces converge”, “star shaped”

  6. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

  7. 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 “evolving hypersurfaces”, “star shaped”

  8. 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”

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) 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)

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.