Paper · Nov 7, 2023

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

Zheng Huang, Longzhi Lin, Zhou Zhang

math.DGarXiv:2311.04298
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Abstract

There has been a conjecture, often attributed to Thurston, which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this paper, for a certain class of almost Fuchsian manifolds, we prove the long-time existence and convergence of the modified mean curvature flow which was first introduced by Xiao and the second named author in. As an application, we confirm Thurston's CMC foliation conjecture for such a subclass of almost Fuchsian manifolds.

Comments: 49 pages

Mean curvature flowGeometrizationLong-time existence & convergence

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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 2021, 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 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)2Huang 2021 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)4de Freitas 2022 Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary (Xinrong Zhao, 2026)8Zhao 2026 Modified mean curvature flow of graphs in Riemannian manifolds (Jocel Faustino Norberto de Oliveira, Jorge Herbert Soares de Lira and Matheus Nunes Soares, 2026)7de Oliveira 2026 Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes (Tanay Gupta, Sukanta Panda and Rajib Saha, 2026)6Gupta 2026 Geometrisation of 3-manifolds (Bruno Martelli, 2026)5Martelli 2026 Modified Mean Curvature Flow in Fuchsian Manifolds (Yuk Shing Lam and Longzhi Lin, 2026)1Lam 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)3Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Modified Mean Curvature Flow in Fuchsian Manifolds

    Yuk Shing Lam and Longzhi LinWider flows PDF

    Topics Long-time existence & convergence · Same author Longzhi Lin · Shared terms “Fuchsian manifold”, “modified mean”, “constant mean”

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

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows PDF

    Topics Long-time existence & convergence · Same authors Zheng Huang, Longzhi Lin, Zhou Zhang

  3. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al. PDF

    Topics Geometrization · Shared terms “Thurston”

  4. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos PDF

    Topics Geometrization · Shared terms “Thurston”

  5. Geometrisation of 3-manifolds

    Bruno Martelli PDF

    Topics Geometrization · Shared terms “Thurston”

  6. Modifications of CMB Temperature and Polarization Quadrupole Signals in Thurston Spacetimes

    Tanay Gupta, Sukanta Panda and Rajib Saha PDF

    Topics Geometrization · Shared terms “Thurston”

  7. Modified mean curvature flow of graphs in Riemannian manifolds

    Jocel Faustino Norberto de Oliveira, Jorge Herbert Soares de Lira and Matheus Nunes SoaresWider flows PDF

    Topics Long-time existence & convergence · Shared terms “modified mean”

  8. Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact -Manifolds with Boundary

    Xinrong Zhao PDF

    Topics Geometrization, Long-time existence & convergence · Shared terms “Thurston”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 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.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.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.