Paper · Sep 19, 2024

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

Yaoting Gui, Yuqiao Li, Jun Sun

math.DGWider flowsarXiv:2409.12810v2
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Abstract

We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau. And also a uniqueness result follows

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Mean curvature flowVolume-preserving & constrained flows3-manifoldsSphere theoremsStability & uniquenessLong-time existence & convergenceMinimal & CMC surfaces

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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)8De Gennaro 2022 The quermassintegral preserving mean curvature flow in the sphere (Esther Cabezas-Rivas and Julian Scheuer, 2022)5Cabezas-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)7De Gennaro 2023 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)6Sinestrari 2025 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)1Gui 2025 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)4Cabezas-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 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 · 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 · Same authors Yaoting Gui, Yuqiao Li, Jun Sun · Shared terms “Schwarzschild space”, “asymptotic schwarzschild”, “curvature surface”

  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, Stability & uniqueness · Shared terms “converges exponentially”, “round sphere”, “flow exists”

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

    Esther Cabezas-Rivas and Carlo SinestrariWider flows PDF

    Topics Sphere theorems, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “space forms”, “sufficiently”, “fundamental form”

  5. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows PDF

    Topics Sphere theorems, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “space forms”, “flow exists”, “Huisken”

  6. 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 “flow exists”, “foliation”, “Huisken”

  7. 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”

  8. 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 “exponentially fast”, “constant mean”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  3. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.