Paper · Nov 30, 2022

The quermassintegral preserving mean curvature flow in the sphere

Esther Cabezas-Rivas, Julian Scheuer

math.DGWider flowsarXiv:2211.17040Analysis & PDE 17 (2024) 3589-3621
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Abstract

We introduce a mean curvature flow with global term of convex hypersurfaces in the sphere, for which the global term can be chosen to keep any quermassintegral fixed. Then, starting from a strictly convex initial hypersurface, we prove that the flow exists for all times and converges smoothly to a geodesic sphere. This provides a workaround to an issue present in the volume preserving mean curvature flow in the sphere introduced by Huisken in 1987. We also classify solutions for some constant curvature type equations in space forms, as well as solitons in the sphere and in the upper branch of the De Sitter space.

Comments: 35 pages

Mean curvature flowVolume-preserving & constrained flowsSphere theoremsLong-time existence & convergenceGeometric inequalitiesPhysics

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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 Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)6Gui 2024 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)7Gui 2025 Constrained Curvature Flows on Pinched Hadamard Surfaces (Sara Albert-Niclòs and Esther Cabezas-Rivas, 2026)8Albert-Niclò 2026 Capillary quermassintegral inequalities in the unit ball (Shujing Pan and Julian Scheuer, 2026)5Pan 2026 The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere (Sara Albert-Niclòs, Esther Cabezas-Rivas and Shujing Pan, 2026)2Albert-Niclò 2026 Geometric C¹-stability of geodesic spheres under volume-preserving mean curvature flow (Esther Cabezas-Rivas and Carlo Sinestrari, 2026)1Cabezas-Riva 2026 Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space (Tianci Luo, Yong Wei, Ruijia Zhang et al., 2026)4Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same author Esther Cabezas-Rivas · Shared terms “geodesic sphere”, “initial hypersurface”, “space forms”

  2. The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

    Sara Albert-Niclòs, Esther Cabezas-Rivas and Shujing PanWider flows PDF

    Topics Geometric inequalities, Long-time existence & convergence · Same author Esther Cabezas-Rivas · Shared terms “quermassintegral preserving”, “convex hypersurfaces”, “geodesic sphere”

  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 “initial hypersurface”, “strictly convex”, “flow exists”

  4. Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows PDF

    Topics Sphere theorems, Geometric inequalities, Long-time existence & convergence · Shared terms “quermassintegral preserving”, “geodesic sphere”, “strictly convex”

  5. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows PDF

    Topics Geometric inequalities · Same author Julian Scheuer · Shared terms “quermassintegral”

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

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

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

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

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

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

  8. Constrained Curvature Flows on Pinched Hadamard Surfaces

    Sara Albert-Niclòs and Esther Cabezas-RivasWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Esther Cabezas-Rivas · Shared terms “constant curvature”, “convex initial”

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) 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) 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. 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. 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.