Paper · May 21, 2016

Mean curvature flow in Fuchsian manifolds

Zheng Huang, Longzhi Lin, Zhou Zhang

math.DGWider flowsarXiv:1605.06565v2
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Abstract

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and the real line. In particular, we prove that there exists a large class of closed initial surfaces, as geodesic graphs over the totally geodesic surface , such that the mean curvature flow exists for all time and converges to . This is among the first examples of converging mean curvature flows of compact hypersurfaces in Riemannian manifolds. We also provide some useful calculations for the general warped product setting.

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Submitted
May 21, 2016
Last revised
May 20, 2019 (v2)
Subjects
math.DG, math.AP, math.GT

Comments: 20 pages, final version, to appear Comm. Cont. Math

Mean curvature flowSingularities3-manifoldsLong-time existence & convergenceRotational symmetry & warped productsMinimal & CMC surfaces

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 Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space (Patrick Allmann, Longzhi Lin and Jingyong Zhu, 2017)5Allmann 2017 Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds (Zheng Huang, Zhou Zhang and Hengyu Zhou, 2017)1Huang 2017 Mean curvature flow in homology and foliations of hyperbolic 3-manifolds (Marco A. M. Guaraco, Vanderson Lima and Franco Vargas Pallete, 2021)8Guaraco 2021 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)4Huang 2021 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)7Fujihara 2023 Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2023)2Huang 2023 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)6Mramor 2024 Modified Mean Curvature Flow in Fuchsian Manifolds (Yuk Shing Lam and Longzhi Lin, 2026)3Lam 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

    Zheng Huang, Zhou Zhang and Hengyu ZhouWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Same authors Zheng Huang, Zhou Zhang · Shared terms “closed initial”, “geodesic graph”, “totally geodesic”

  2. Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds

    Zheng Huang, Longzhi Lin and Zhou Zhang

    Topics Long-time existence & convergence · Same authors Zheng Huang, Longzhi Lin, Zhou Zhang · Shared terms “Fuchsian manifolds”

  3. Modified Mean Curvature Flow in Fuchsian Manifolds

    Yuk Shing Lam and Longzhi LinWider flows

    Topics Long-time existence & convergence · Same author Longzhi Lin · Shared terms “Fuchsian manifolds”

  4. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Long-time existence & convergence · Same authors Zheng Huang, Longzhi Lin, Zhou Zhang · Shared terms “flow exists”

  5. Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space

    Patrick Allmann, Longzhi Lin and Jingyong ZhuWider flows

    Topics Minimal & CMC surfaces · Same author Longzhi Lin · Shared terms “geometric flows”

  6. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics Minimal & CMC surfaces, 3-manifolds, Singularities · Shared terms “finite time”

  7. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “geodesic graph”, “Riemannian manifold”

  8. Mean curvature flow in homology and foliations of hyperbolic -manifolds

    Marco A. M. Guaraco, Vanderson Lima and Franco Vargas PalleteWider flows

    Topics Minimal & CMC surfaces, Long-time existence & convergence · Shared terms “Fuchsian”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) 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. 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.