Paper · Dec 16, 2017

Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

Kunbo Wang

math.DGWider flowsarXiv:1712.05918
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As on arXiv

Abstract

In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric -dimensional hypersurface in between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.

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Submitted
Dec 16, 2017
Subjects
math.DG
DOI
10.1007/s11425-000-0000-0

Comments: 10 pages

Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryLong-time existence & convergenceRotational symmetry & warped productsMinimal & CMC surfaces

Its neighbourhood

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 2016, 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 Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces (Valentina-Mira Wheeler, 2016)3Wheeler 2016 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)4Wheeler 2016 Singularities of the area preserving curve shortening flow with a Neumann free boundary condition (Elena Mäder-Baumdicker, 2017)7Mäder-Baumdi 2017 Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball (Guofang Wang and Chao Xia, 2019)1Wang 2019 Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces (Brian Harvie, 2020)8Harvie 2020 Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)2Gui 2024 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)5Gui 2025 Area-preserving curvature flow for open curves with general contact angles on skew lines (Fuya Hiroi, 2026)6Hiroi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball

    Guofang Wang and Chao XiaWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence

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

    Yaoting Gui, Yuqiao Li and Jun SunWider flows

    Topics Minimal & CMC surfaces, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “area preserving”, “constant mean”

  3. Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

    Valentina-Mira WheelerWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary, Long-time existence & convergence

  4. Mean curvature flow with free boundary - Type 2 singularities

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Rotational symmetry & warped products · Shared terms “rotationally symmetric”

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

    Yaoting Gui, Yuqiao Li and Jun SunWider flows

    Topics Minimal & CMC surfaces, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “area preserving”, “constant mean”

  6. Area-preserving curvature flow for open curves with general contact angles on skew lines

    Fuya HiroiWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “area preserving”

  7. Singularities of the area preserving curve shortening flow with a Neumann free boundary condition

    Elena Mäder-BaumdickerWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “area preserving”, “Neumann”

  8. Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces

    Brian HarvieWider flows

    Topics Minimal & CMC surfaces, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “symmetric hypersurfaces”, “rotationally symmetric”

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 in the L² sense. 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 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)

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.