Paper · Dec 22, 2024

Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces

Yaoting Gui, Yuqiao Li, Jun Sun

math.DGWider flowsarXiv:2412.17024v2
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Abstract

This paper investigates the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces. We demonstrate the long-time existence and exponential convergence of this flow with a coordinate sphere of large radius serving as the initial surface in the asymptotically flat end, which eventually converges to a constant harmonic mean curvature surface. We also establish that these surfaces form a foliation of the space outside a large ball. Finally, we utilize this foliation to define the center of mass, proving that it agrees with the center of mass defined by the ADM formulation of the initial data set.

Comments: Comments are welcome, a missing term is added in Lemma 2.2, some formulas are modified

Mean curvature flowBoundaries & non-compactLong-time existence & convergencePhysics

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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 A volume-renormalized mass for asymptotically hyperbolic manifolds (Mattias Dahl, Klaus Kroencke and Stephen McCormick, 2023)8Dahl 2023 Mean Curvature Flow in de Sitter space (Or Hershkovits and Leonardo Senatore, 2023)6Hershkovits 2023 Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds (Luen-Fai Tam, 2024)7Tam 2024 Volume preserving spacetime mean curvature flow and foliations of initial data sets (Jacopo Tenan, 2024)4Tenan 2024 Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)2Gui 2024 Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces (Carlo Sinestrari and Jacopo Tenan, 2025)5Sinestrari 2025 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)1Gui 2025 Foliations by constant spacetime mean curvature surfaces for asymptotically hyperboloidal initial data sets (Jacopo Tenan, 2026)3Tenan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Long-time existence & convergence, Boundaries & non-compact · Same authors Yaoting Gui, Yuqiao Li, Jun Sun · Shared terms “Schwarzschild space”, “asymptotic schwarzschild”, “coordinate sphere”

  2. Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Long-time existence & convergence · Same authors Yaoting Gui, Yuqiao Li, Jun Sun · Shared terms “Schwarzschild space”, “asymptotic schwarzschild”, “curvature surface”

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

    Jacopo TenanWider flows PDF

    Topics Physics · Shared terms “curvature surface”, “data set”, “foliation”

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    Jacopo TenanWider flows PDF

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

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

    Carlo Sinestrari and Jacopo TenanWider flows PDF

    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Shared terms “asymptotically flat”, “volume preserving”, “initial surface”

  6. Mean Curvature Flow in de Sitter space

    Or Hershkovits and Leonardo SenatoreWider flows PDF

    Topics Physics, Long-time existence & convergence

  7. Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

    Luen-Fai TamWider flows PDF

    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Schwarzschild”, “asymptotically flat”

  8. A volume-renormalized mass for asymptotically hyperbolic manifolds

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    Topics Physics, Boundaries & non-compact · Shared terms “ADM”

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) 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) 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.