Paper · Apr 9, 2020

Long time behaviour of discrete volume preserving mean curvature flows

Massimiliano Morini, Marcello Ponsiglione, Emanuele Spadaro

math.APWider flowsarXiv:2004.04799
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Abstract

In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.

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Submitted
Apr 9, 2020
Subjects
math.AP
Mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergence

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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 Volume preserving mean curvature flows near strictly stable sets in flat torus (Joonas Niinikoski, 2019)7Niinikoski 2019 Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2020)5Julin 2020 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)4Huang 2021 The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions (Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al., 2021)1Julin 2021 Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus (Daniele De Gennaro and Anna Kubin, 2022)2De Gennaro 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)8De Gennaro 2023 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)3Bonforte 2024 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)6Arya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows

    Topics Volume-preserving & constrained flows · Same authors Massimiliano Morini, Marcello Ponsiglione, Emanuele Spadaro · Shared terms “finite union”, “finite perimeter”, “disjoint”

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

    Daniele De Gennaro and Anna KubinWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “discrete volume”, “time behaviour”, “finite perimeter”

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

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows

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

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

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

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

  5. Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Volume-preserving & constrained flows · Shared terms “disjoint balls”, “finite perimeter”

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

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “disjoint balls”, “discrete”, “perimeter”

  7. Volume preserving mean curvature flows near strictly stable sets in flat torus

    Joonas NiinikoskiWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “initial set”

  8. 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

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

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) 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) 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.