Paper · Jul 26, 2024

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

Matteo Bonforte, Francesco Maggi, Daniel Restrepo

math.APWider flowsarXiv:2407.18868
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As on arXiv

Abstract

We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused balls".

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Submitted
Jul 26, 2024
Subjects
math.AP, math.DG, math.OC

Comments: 51 pages

Mean curvature flowVolume-preserving & constrained flowsLong-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)1Huang 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)5Laux 2022 Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications (Ben Lambert and Elena Mäder-Baumdicker, 2022)4Lambert 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)2De Gennaro 2023 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)8Jang 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)7Chiesa 2025 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)3Arya 2026 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)6Cabezas-Riva 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 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 · Shared terms “Euclidean space”

  2. 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

  3. 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

  4. Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications

    Ben Lambert and Elena Mäder-BaumdickerWider flows

    Topics Volume-preserving & constrained flows · Shared terms “Euclidean”

  5. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows

    Topics Volume-preserving & constrained flows

  6. Geometric -stability of geodesic spheres under volume-preserving mean curvature flow

    Esther Cabezas-Rivas and Carlo SinestrariWider flows

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

  7. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows

    Topics Volume-preserving & constrained flows

  8. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    Jiwoong JangWider flows

    Topics Volume-preserving & constrained flows, 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) 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.