Paper · Mar 11, 2019

Scherk-like Translators for Mean Curvature Flow

David Hoffman, Francisco Martín, Brian White

math.DGWider flowsarXiv:1903.04617v4
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As on arXiv

Abstract

We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.

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Submitted
Mar 11, 2019
Last revised
Feb 4, 2020 (v4)
Subjects
math.DG, math.AP

Comments: 43 pages, 14 Figures. This revised version will appear in Journal of Differential Geometry

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessMinimal & 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 Graphical Translators for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martin et al., 2018)2Hoffman 2018 Nguyen's Tridents and the Classification of Semigraphical Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)3Hoffman 2019 Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries (Brian White, 2021)7White 2021 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)4Gama 2022 Annuloids and Δ-wings (D. Hoffman, F. Martin and B. White, 2023)6Hoffman 2023 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)5Hoffman 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)8Martín 2023 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)1Martín 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of Semigraphical Translators

    Francisco Martín, Mariel Sáez, Raphael Tsiamis et al.Wider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same authors Francisco Martín, Brian White · Shared terms “helicoid”, “pitchfork”

  2. Graphical Translators for Mean Curvature Flow

    David Hoffman, Tom Ilmanen, Francisco Martin et al.Wider flows

    Topics Self-shrinkers & translators · Same authors David Hoffman, Francisco Martín, Brian White · Shared terms “parameter family”

  3. Nguyen's Tridents and the Classification of Semigraphical Translators for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Self-shrinkers & translators · Same authors David Hoffman, Francisco Martín, Brian White · Shared terms “parameter family”, “periodic”

  4. Finite entropy translating solitons in slabs

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Stability & uniqueness · Same author Francisco Martín · Shared terms “pitchfork”

  5. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Self-shrinkers & translators · Same authors David Hoffman, Francisco Martín, Brian White

  6. Annuloids and -wings

    D. Hoffman, F. Martin and B. WhiteWider flows

    Topics Self-shrinkers & translators · Same authors David Hoffman, Francisco Martín, Brian White

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

    Brian White

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

  8. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Francisco Martín · Shared terms “helicoid”, “pitchfork”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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)

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