Paper · Sep 15, 2016

Mean curvature flow with free boundary - Type 2 singularities

Glen Wheeler, Valentina-Mira Wheeler

math.DGWider flowsarXiv:1609.04494
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Abstract

In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities may only occur at infinity.

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Submitted
Sep 15, 2016
Subjects
math.DG

Comments: 12 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1605.09098

Mean curvature flowOther curvature flowsSingularitiesFree 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 Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces (Valentina-Mira Wheeler, 2016)1Wheeler 2016 Concentration-compactness and finite-time singularities for Chen's flow (Yann Bernard, Glen Wheeler and Valentina-Mira Wheeler, 2017)6Bernard 2017 On a curvature flow model for embryonic epidermal wound healing (Shuhui He, Glen Wheeler and Valentina-Mira Wheeler, 2018)3He 2018 Rotational symmetry of solutions of mean curvature flow coming out of a double cone (Letian Chen, 2021)8Chen 2021 Mean curvature flow of symmetric double graphs only develops singularities on the hyperplane of symmetry (Wolfgang Maurer, 2021)5Maurer 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)4Chen 2022 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)2Chai 2022 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)7Ghosh 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 in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

    Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Singularities · Same author Valentina-Mira Wheeler

  2. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary, Other curvature flows, Singularities

  3. On a curvature flow model for embryonic epidermal wound healing

    Shuhui He, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Other curvature flows, Singularities · Same authors Glen Wheeler, Valentina-Mira Wheeler

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

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Singularities · Shared terms “double cone”, “symmetric double”, “rotationally symmetric”

  5. 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 “symmetric double”

  6. Concentration-compactness and finite-time singularities for Chen's flow

    Yann Bernard, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Other curvature flows, Singularities · Same authors Glen Wheeler, Valentina-Mira Wheeler

  7. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows

    Topics Free boundary & capillary, Singularities

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

    Letian ChenWider flows

    Topics Rotational symmetry & warped products · Shared terms “double cone”, “symmetric double”, “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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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)

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  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.