Paper · Jul 17, 2024

Volume preserving spacetime mean curvature flow and foliations of initial data sets

Jacopo Tenan

math.DGWider flowsarXiv:2407.12971v4
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Abstract

We consider a volume preserving curvature evolution of surfaces in an asymptotically Euclidean initial data set with positive ADM-energy. The speed is given by a nonlinear function of the mean curvature which generalizes the spacetime mean curvature recently considered by Cederbaum-Sakovich (Calc. Var. PDE, 2021). Following a classical approach by Huisken-Yau (Invent. Math., 1996), we show that the flow starting from suitably round initial surfaces exists for all times and converges to a constant (spacetime) curvature limit. This provides an alternative construction of the CSTMC foliation by Cederbaum-Sakovich and has applications in the definition of center of mass of an isolated system in General Relativity.

Comments: 33 pages

Mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergencePhysics

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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)4Huang 2021 Hawking Mass Monotonicity for Initial Data Sets (Sven Hirsch, 2022)7Hirsch 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)5Bonforte 2024 Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)3Gui 2024 Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (Klaus Kroencke and Markus Wolff, 2024)6Kroencke 2024 Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces (Carlo Sinestrari and Jacopo Tenan, 2025)2Sinestrari 2025 Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets (Jacopo Tenan, 2026)1Tenan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “spacetime mean”, “data set”, “preserving spacetime”

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

    Carlo Sinestrari and 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”

  3. Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Physics, Long-time existence & convergence · Shared terms “data set”, “foliation”, “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 · Shared terms “Huisken”, “Euclidean”

  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 · Shared terms “Euclidean”

  6. Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature

    Klaus Kroencke and Markus WolffWider flows PDF

    Topics Long-time existence & convergence · Shared terms “spacetime mean”, “foliation”, “Huisken Yau”

  7. Hawking Mass Monotonicity for Initial Data Sets

    Sven HirschWider flows PDF

    Topics Physics · Shared terms “data set”, “initial data”, “foliation”

  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 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) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.