Paper · May 18, 2023

Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

Daniele De Gennaro, Antonia Diana, Andrea Kubin, Anna Kubin

math.DGWider flowsarXiv:2305.11100v2
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Abstract

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting close to a strictly stable critical set of the perimeter , exist for all times and converge to a translate of exponentially fast as time goes to infinity.

Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessLong-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 Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus (Daniele De Gennaro and Anna Kubin, 2022)1De Gennaro 2022 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)4Daniele 2022 Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles (Marco Cicalese and Andrea Kubin, 2024)8Cicalese 2024 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)5Bonforte 2024 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 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)6Kubin 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. Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

    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”, “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”, “exponentially fast”

  3. 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, Stability & uniqueness

  4. 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 authors Andrea Kubin, Anna Kubin · Shared terms “volume preserving”, “exponentially fast”, “perimeter”

  5. 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

  6. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Andrea Kubin

  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. Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

    Marco Cicalese and Andrea KubinWider flows PDF

    Topics Volume-preserving & constrained flows, Stability & uniqueness · Same author Andrea Kubin

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.