Paper · Apr 5, 2023

A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

Wilhelm Klingenberg, Ben Lambert, Julian Scheuer

math.DGWider flowsarXiv:2304.02414J. Evol. Equ. 25 (2025), no. 1, art. 15
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Abstract

Consider a convex cone in three-dimensional Minkowski space which either contains the lightcone or is contained in it. This work considers mean curvature flow of a proper spacelike strictly mean convex disc in the cone which is graphical with respect to its rays. Its boundary is required to have constant intersection angle with the boundary of the cone. We prove that the corresponding parabolic boundary value problem for the graph admits a solution for all time which rescales to a self-similarly expanding solution.

Comments: 19 pages

Mean curvature flow3-manifoldsFree boundary & capillaryPhysics

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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 Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)3Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ℝ²₁ (Ya Gao, Chenyang Liu and Jing Mao, 2021)8Gao 2021 Entire self-expanders for power of σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)7Wang 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)4Wang 2022 Parabolic evolution with boundary to the Bishop family of holomorphic discs (Brendan Guilfoyle and Wilhelm Klingenberg, 2023)1Guilfoyle 2023 On the second boundary value problem for mean curvature flow in Minkowski space (Rongli Huang, 2024)6Huang 2024 Capillary quermassintegral inequalities in the unit ball (Shujing Pan and Julian Scheuer, 2026)2Pan 2026 Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity (Dake Li, Zhizhang Wang and Shiqi Yin, 2026)5Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Parabolic evolution with boundary to the Bishop family of holomorphic discs

    Brendan Guilfoyle and Wilhelm KlingenbergWider flows PDF

    Topics Free boundary & capillary · Same author Wilhelm Klingenberg · Shared terms “required”, “graphical”

  2. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows PDF

    Topics Free boundary & capillary · Same author Julian Scheuer

  3. Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Physics · Shared terms “Minkowski space”, “spacelike”

  4. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows PDF

    Topics Physics · Shared terms “Minkowski space”, “spacelike”

  5. Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity

    Dake Li, Zhizhang Wang and Shiqi YinWider flows PDF

    Topics Physics · Shared terms “Minkowski space”, “spacelike strictly”, “boundary value”

  6. On the second boundary value problem for mean curvature flow in Minkowski space

    Rongli HuangWider flows PDF

    Topics Mean curvature flow · Shared terms “Minkowski space”, “boundary value”

  7. Entire self-expanders for power of curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows PDF

    Topics Physics · Shared terms “Minkowski space”, “spacelike”

  8. An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

    Ya Gao, Chenyang Liu and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Physics · Shared terms “spacelike”, “Minkowski”

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) 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) 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) 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. 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.

  2. 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.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.