Paper · Jan 22, 2025

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

Carlo Sinestrari, Jacopo Tenan

math.DGWider flowsarXiv:2501.13091
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Abstract

We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat -manifold modeling an isolated gravitating system in General Relativity. We show that, if the ambient manifold has positive ADM mass and the initial surface is round in a suitable sense, then the flow exists for all times and converges smoothly to a stable CMC surface. This extends to the asymptotically flat setting a classical result by Huisken-Yau (Invent. Math. 1996) and allows to construct a CMC foliation of the outer part of the manifold by an alternative approach to the ones by Nerz (Calc. Var. PDE, 2015) or by Eichmair-Koerber (J. Diff. Geometry, 2024).

Comments: 29 pages

Mean curvature flowVolume-preserving & constrained flowsBoundaries & non-compactLong-time existence & convergenceMinimal & CMC surfacesPhysics

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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)3Huang 2021 Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus (Daniele De Gennaro and Anna Kubin, 2022)7De Gennaro 2022 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)4Sinestrari 2024 Volume preserving spacetime mean curvature flow and foliations of initial data sets (Jacopo Tenan, 2024)1Tenan 2024 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)8Bonforte 2024 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)2Gui 2025 Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets (Jacopo Tenan, 2026)6Tenan 2026 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)5Cabezas-Riva 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume preserving spacetime mean curvature flow and foliations of initial data sets

    Jacopo TenanWider flows PDF

    Topics Volume-preserving & constrained flows, Physics, Long-time existence & convergence · Same author Jacopo Tenan · Shared terms “positive ADM”, “Huisken Yau”, “initial surface”

  2. 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 “flat space”, “asymptotically flat”, “converges smoothly”

  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 · Shared terms “flow exists”, “Huisken”

  4. Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

    Carlo Sinestrari and Liangjun WengWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Carlo Sinestrari · Shared terms “flow exists”

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

    Esther Cabezas-Rivas and Carlo SinestrariWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Carlo Sinestrari

  6. Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets

    Jacopo TenanWider flows PDF

    Topics Physics · Same author Jacopo Tenan · Shared terms “volume preserving”, “foliation”, “Calc”

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

    Daniele De Gennaro and Anna KubinWider flows PDF

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

  8. 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

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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.