Paper · Nov 1, 2025

Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space

Yaoting Gui, Yuqiao Li, Jun Sun

math.DGWider flowsarXiv:2511.00435v2
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Abstract

In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.

Comments: Comments are welcome

Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessBoundaries & non-compactLong-time existence & convergenceGeometric inequalitiesMinimal & CMC surfaces

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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 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 The quermassintegral preserving mean curvature flow in the sphere (Esther Cabezas-Rivas and Julian Scheuer, 2022)6Cabezas-Riva 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)5De Gennaro 2023 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)8Bonforte 2024 Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)1Gui 2024 Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces (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)3Sinestrari 2025
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 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 · Same authors Yaoting Gui, Yuqiao Li, Jun Sun · Shared terms “Schwarzschild space”, “asymptotic schwarzschild”, “area preserving”

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

    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”

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

    Carlo Sinestrari and Jacopo TenanWider flows PDF

    Topics Minimal & CMC surfaces, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “flat space”, “asymptotically flat”, “converges smoothly”

  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, Stability & uniqueness · Shared terms “initial hypersurface”, “flow exists”

  5. 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, Stability & uniqueness · Shared terms “exponentially fast”

  6. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows PDF

    Topics Geometric inequalities, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “initial hypersurface”, “converges smoothly”, “flow exists”

  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 · Shared terms “constant mean”, “exponentially fast”

  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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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)

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.