Paper · Jul 22, 2026

Modified Mean Curvature Flow in Fuchsian Manifolds

Yuk Shing Lam, Longzhi Lin

math.DGWider flowsarXiv:2607.20350
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Abstract

In this paper, we show that the modified mean curvature flow starting from an arbitrary graph in a Fuchsian manifold exists for all time and converges smoothly to an equidistant surface of constant mean curvature as . This result generalizes earlier work to the modified mean curvature flow setting and removes the restrictive global gradient bound initially required for the standard mean curvature flow by Huang, Zhou, and the second author.

Comments: 38 pages. Comments welcome!

Mean curvature flowLong-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)3Huang 2021 A local estimate for the mean curvature flow (Zhen Wang, 2021)7Wang 2021 Mean Curvature Flow in de Sitter space (Or Hershkovits and Leonardo Senatore, 2023)4Hershkovits 2023 Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2023)1Huang 2023 Mean curvature flow with multiplicity 2 convergence in ℝ³ (Jingwen Chen and Ao Sun, 2023)5Chen 2023 Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (Klaus Kroencke and Markus Wolff, 2024)8Kroencke 2024 On the rate of convergence of cylindrical singularity in mean curvature flow (Yiqi Huang and Xinrui Zhao, 2025)6Huang 2025 Modified mean curvature flow of graphs in Riemannian manifolds (Jocel Faustino Norberto de Oliveira, Jorge Herbert Soares de Lira and Matheus Nunes Soares, 2026)2de Oliveira 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 and CMC foliation conjecture in almost Fuchsian manifolds

    Zheng Huang, Longzhi Lin and Zhou Zhang PDF

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

  2. 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”

  3. 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 author Longzhi Lin

  4. Mean Curvature Flow in de Sitter space

    Or Hershkovits and Leonardo SenatoreWider flows PDF

    Topics Long-time existence & convergence · Shared terms “converges smoothly”

  5. Mean curvature flow with multiplicity 2 convergence in

    Jingwen Chen and Ao SunWider flows PDF

    Topics Long-time existence & convergence

  6. On the rate of convergence of cylindrical singularity in mean curvature flow

    Yiqi Huang and Xinrui ZhaoWider flows PDF

    Topics Long-time existence & convergence

  7. A local estimate for the mean curvature flow

    Zhen WangWider flows PDF

    Topics Long-time existence & convergence

  8. Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature

    Klaus Kroencke and Markus WolffWider flows PDF

    Topics Long-time existence & convergence

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) 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.