Paper · Mar 10, 2021

Mean curvature flow of symmetric double graphs only develops singularities on the hyperplane of symmetry

Wolfgang Maurer

math.DGWider flowsarXiv:2103.06072
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Abstract

By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these properties and singularities only occur on the hyperplane of symmetry. The result can be used to construct smooth solutions to the free Neumann boundary problem on a supporting hyperplane with singular boundary. For the construction we introduce and investigate a notion named "vanity" and which is similar to convexity. Moreover, we rely on Sáez' and Schnürer's "mean curvature flow without singularities" to approximate weak solutions with smooth graphical solutions in one dimension higher.

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Submitted
Mar 10, 2021
Subjects
math.DG
Mean curvature flowSingularitiesWeak & singular flowsFree boundary & capillary

Its neighbourhood

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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 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 Shadows of graphical mean curvature flow (Wolfgang Maurer, 2016)3Maurer 2016 Mean Curvature Flows of Graphs with Neumann Boundary condition (Jinju Xu, 2016)2Xu 2016 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)1Wheeler 2016 Minimizing movements for mean curvature flow of droplets with prescribed contact angle (Giovanni Bellettini and Shokhrukh Yusufovich Kholmatov, 2016)5Bellettini 2016 Singularity formation in axially symmetric mean curvature flow with Neumann boundary (John Head and Sevvandi Kandanaarachchi, 2019)8Head 2019 Level set mean curvature flow with Neumann boundary conditions (Satoru Aimi, 2021)4Aimi 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)6Chen 2022 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)7Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow with free boundary - Type 2 singularities

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Singularities · Shared terms “symmetric double”

  2. Mean Curvature Flows of Graphs with Neumann Boundary condition

    Jinju XuWider flows

    Topics Free boundary & capillary · Shared terms “Neumann boundary”

  3. Shadows of graphical mean curvature flow

    Wolfgang MaurerWider flows

    Topics Weak & singular flows · Same author Wolfgang Maurer · Shared terms “graphical”

  4. Level set mean curvature flow with Neumann boundary conditions

    Satoru AimiWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “Neumann boundary”

  5. Minimizing movements for mean curvature flow of droplets with prescribed contact angle

    Giovanni Bellettini and Shokhrukh Yusufovich KholmatovWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “hyperplane”

  6. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Weak & singular flows, Singularities · Shared terms “symmetric double”

  7. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Weak & singular flows, Singularities

  8. Singularity formation in axially symmetric mean curvature flow with Neumann boundary

    John Head and Sevvandi KandanaarachchiWider flows

    Topics Free boundary & capillary, Singularities · Shared terms “Neumann boundary”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.