Paper · May 16, 2021

Mean curvature flow in homology and foliations of hyperbolic -manifolds

Marco A. M. Guaraco, Vanderson Lima, Franco Vargas Pallete

math.DGWider flowsarXiv:2105.07504
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Abstract

We study global aspects of the mean curvature flow of non-separating hypersurfaces in closed manifolds. For instance, if has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal hypersurface . We prove a similar result for the flow with surgery in dimension 2. As an application we show the existence of monotone incompressible isotopies in manifolds with negative curvature. Combining this result with min-max theory, we show that quasi-Fuchsian and hyperbolic -manifolds fibered over admit smooth entire foliations whose leaves are either minimal or have non-vanishing mean curvature. We also conclude the existence of outermost minimal surfaces for quasi-Fuchsian ends and study their continuity with respect to variations of the quasi-Fuchsian metric.

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Submitted
May 16, 2021
Subjects
math.DG, math.GT

Comments: 31 pages, 7 figures

Mean curvature flowSurgeryWeak & singular flowsLong-time existence & convergenceMinimal & CMC surfaces

Its neighbourhood

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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 Mean curvature flow in Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2016)6Huang 2016 Minimal two-spheres in three-spheres (Robert Haslhofer and Daniel Ketover, 2017)7Haslhofer 2017 Regularity and stability results for the level set flow via the mean curvature flow with surgery (Alexander Mramor, 2017)1Mramor 2017 Free boundary flow with surgery (Robert Haslhofer, 2023)4Haslhofer 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)2Chen 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)5Chu 2024 On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume (Ruojing Jiang and Franco Vargas Pallete, 2025)3Jiang 2025 Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds (Ruojing Jiang and Franco Vargas Pallete, 2025)8Jiang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Regularity and stability results for the level set flow via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Surgery, Weak & singular flows · Shared terms “set flow”, “level set”

  2. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows

    Topics Minimal & CMC surfaces, Long-time existence & convergence · Shared terms “closed manifold”, “minimal hypersurfaces”

  3. On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

    Ruojing Jiang and Franco Vargas Pallete

    Topics Minimal & CMC surfaces, Long-time existence & convergence · Same author Franco Vargas Pallete

  4. Free boundary flow with surgery

    Robert HaslhoferWider flows

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

  5. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows

    Topics Minimal & CMC surfaces · Shared terms “embedded minimal”, “max theory”, “min max”

  6. Mean curvature flow in Fuchsian manifolds

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

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

  7. Minimal two-spheres in three-spheres

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces · Shared terms “embedded minimal”, “max theory”, “min max”

  8. Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

    Ruojing Jiang and Franco Vargas Pallete

    Topics Minimal & CMC surfaces · Same author Franco Vargas Pallete

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  3. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.