Paper · Jan 11, 2022

Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

Daniele De Gennaro, Anna Kubin

math.APWider flowsarXiv:2201.04174v2
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Abstract

We show that the discrete approximate volume preserving mean curvature flow in the flat torus starting near a strictly stable critical set of the perimeter converges in the long time to a translate of exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.

Comments: 38 pages, 2 figure. arXiv admin note: text overlap with arXiv:2004.04799 by other authors

Mean curvature flowVolume-preserving & constrained flowsSurfacesLong-time existence & convergenceMinimal & CMC surfaces

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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 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)4Huang 2021 Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem (De Gennaro Daniele, Andrea Kubin and Anna Kubin, 2022)3Daniele 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)1De Gennaro 2023 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)6Bonforte 2024 Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces (Carlo Sinestrari and Jacopo Tenan, 2025)5Sinestrari 2025 The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus (Vedansh Arya, Daniele De Gennaro and Anna Kubin, 2025)2Arya 2025 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)8Gui 2025 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)7Arya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same authors Daniele De Gennaro, Anna Kubin · Shared terms “flat torus”, “strictly stable”, “stable critical”

  2. The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus

    Vedansh Arya, Daniele De Gennaro and Anna KubinWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same authors Daniele De Gennaro, Anna Kubin · Shared terms “flat torus”, “periodic”, “exponentially fast”

  3. Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

    De Gennaro Daniele, Andrea Kubin and Anna KubinWider flows PDF

    Topics Long-time existence & convergence · Same author Anna Kubin · Shared terms “discrete volume”, “time behaviour”, “volume preserving”

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

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence

  5. Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces

    Carlo Sinestrari and Jacopo TenanWider flows PDF

    Topics Minimal & CMC surfaces, Volume-preserving & constrained flows, Long-time existence & convergence

  6. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence

  7. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “exponentially fast”, “perimeter”

  8. Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Minimal & CMC surfaces, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “constant mean”, “exponentially fast”

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.