Paper · May 30, 2016

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

Valentina-Mira Wheeler

math.DGWider flowsarXiv:1605.09098v2
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Abstract

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.

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Submitted
May 30, 2016
Last revised
Sep 15, 2016 (v2)
Subjects
math.DG

Comments: 23 pages

Mean curvature flowSingularitiesStability & uniquenessFree boundary & capillaryLong-time existence & convergenceMinimal & 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 - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)1Wheeler 2016 Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries (Kunbo Wang, 2017)7Wang 2017 Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders (Katherine Castro and César Rosales, 2022)5Castro 2022 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)8Chai 2022 Free boundary flow with surgery (Robert Haslhofer, 2023)4Haslhofer 2023 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)3Haslhofer 2023 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)2Bourni 2026 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)6Ghosh 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 · Same author Valentina-Mira Wheeler

  2. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary, Long-time existence & convergence · Shared terms “minimal hypersurfaces”

  3. Free boundary minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary · Shared terms “minimal disk”

  4. Free boundary flow with surgery

    Robert HaslhoferWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary, Long-time existence & convergence · Shared terms “minimal disk”, “finite time”

  5. Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

    Katherine Castro and César RosalesWider flows

    Topics Minimal & CMC surfaces, Singularities, Stability & uniqueness · Shared terms “minimal hypersurfaces”

  6. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows

    Topics Free boundary & capillary, Singularities, Stability & uniqueness

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

    Kunbo WangWider flows

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

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

    Xiaoxiang ChaiWider flows

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

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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

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    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.