Paper · Aug 7, 2019

Singularity formation in axially symmetric mean curvature flow with Neumann boundary

John Head, Sevvandi Kandanaarachchi

math.DGWider flowsarXiv:1908.02871
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As on arXiv

Abstract

We study mean curvature flow of smooth, axially symmetric surfaces in with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

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Submitted
Aug 7, 2019
Subjects
math.DG, math.AP
Mean curvature flowSingularitiesFree boundary & capillaryRotational symmetry & warped products

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 Neumann boundary value problem for gernera curvature flow with forcing term (Ling Xiao, 2016)6Xiao 2016 Mean Curvature Flows of Graphs with Neumann Boundary condition (Jinju Xu, 2016)1Xu 2016 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)7Wheeler 2016 Contracting axially symmetric hypersurfaces by powers of the σₖ-curvature (Haizhong Li, Xianfeng Wang and Jing Wu, 2019)4Li 2019 Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold Mⁿ×ℝ (Ya Gao, Yi-Juan Gong and Jing Mao, 2020)2Gao 2020 Mean curvature flow of symmetric double graphs only develops singularities on the hyperplane of symmetry (Wolfgang Maurer, 2021)8Maurer 2021 Level set mean curvature flow with Neumann boundary conditions (Satoru Aimi, 2021)3Aimi 2021 Parabolic evolution with boundary to the Bishop family of holomorphic discs (Brendan Guilfoyle and Wilhelm Klingenberg, 2023)5Guilfoyle 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Flows of Graphs with Neumann Boundary condition

    Jinju XuWider flows

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

  2. Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold

    Ya Gao, Yi-Juan Gong and Jing MaoWider flows

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

  3. Level set mean curvature flow with Neumann boundary conditions

    Satoru AimiWider flows

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

  4. Contracting axially symmetric hypersurfaces by powers of the -curvature

    Haizhong Li, Xianfeng Wang and Jing WuWider flows

    Topics Rotational symmetry & warped products · Shared terms “axially symmetric”

  5. Parabolic evolution with boundary to the Bishop family of holomorphic discs

    Brendan Guilfoyle and Wilhelm KlingenbergWider flows

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

  6. Neumann boundary value problem for gernera curvature flow with forcing term

    Ling XiaoWider flows

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

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

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Rotational symmetry & warped products, Singularities

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

    Wolfgang MaurerWider 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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 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.

  2. 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.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.