Paper · Jun 13, 2021

Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries

Brian White

math.DGarXiv:2106.06893v4
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Abstract

For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that . In this paper, we use mean curvature flow to show that . We get a slightly larger lower bound for orientable surfaces.

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Submitted
Jun 13, 2021
Last revised
Aug 10, 2022 (v4)
Subjects
math.DG
Journal
Comment. Math. Helv.97(2022), no.4, 669-689

Comments: 18 pages. The new version (August 10, 2022) has some added explanatory material (at the suggestion of the CMH referees) and has an updated bibliography

Mean curvature flowRicci solitonsSingularitiesMinimal & 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 Analysis of Velacuteazquez's solution to the mean curvature flow with a type II singularity (Siao-Hao Guo and Natasa Sesum, 2017)6Guo 2017 Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II (Nikolaos Kapouleas and Peter McGrath, 2017)4Kapouleas 2017 Scherk-like Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)1Hoffman 2019 On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds (Maxwell Stolarski, 2023)5Stolarski 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)7Choi 2024 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)2Ilmanen 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)8Mramor 2024 Generating Shrinkers by Mean Curvature Flow (David Hoffman, Francisco Martin and Brian White, 2025)3Hoffman 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Scherk-like Translators for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Minimal & CMC surfaces, Singularities · Same author Brian White

  2. Fattening in mean curvature flow

    Tom Ilmanen and Brian White

    Topics Singularities, Ricci solitons · Same author Brian White · Shared terms “smoothly embedded”, “following property”, “tangent flow”

  3. Generating Shrinkers by Mean Curvature Flow

    David Hoffman, Francisco Martin and Brian White

    Topics Singularities, Ricci solitons · Same author Brian White · Shared terms “smoothly embedded”

  4. Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II

    Nikolaos Kapouleas and Peter McGrath

    Topics Minimal & CMC surfaces, Singularities, Ricci solitons · Shared terms “open set”

  5. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows

    Topics Minimal & CMC surfaces, Singularities · Shared terms “tangent flow”

  6. Analysis of Velzquez's solution to the mean curvature flow with a type singularity

    Siao-Hao Guo and Natasa SesumWider flows

    Topics Minimal & CMC surfaces, Singularities

  7. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Minimal & CMC surfaces · Shared terms “total curvature”

  8. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics Minimal & CMC surfaces, Singularities

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.