Paper · Sep 19, 2019

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

David Hoffman, Francisco Martín, Brian White

math.DGWider flowsarXiv:1909.09241v2
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Abstract

We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection of vertical lines, they are graphs. We also provide a nearly complete classification of semigraphical translators.

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Submitted
Sep 19, 2019
Last revised
Feb 1, 2022 (v2)
Subjects
math.DG, math.AP

Comments: 26 pages, 5 Figures. J. Reine Angew. Math., to appear

Mean curvature flowSelf-shrinkers & translators

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 Translating solitons of the mean curvature flow asymptotic to hyperplanes in ℝⁿ⁺¹ (Eddygledson S. Gama and Francisco Martin, 2018)8Gama 2018 Graphical Translators for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martin et al., 2018)4Hoffman 2018 Notes on translating solitons for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martín et al., 2019)7Hoffman 2019 Scherk-like Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)5Hoffman 2019 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)3Hoffman 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)2Martín 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 · Same authors Francisco Martín, Brian White · Shared terms “semigraphical translators”

  2. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows

    Topics Self-shrinkers & translators · Same author Francisco Martín · Shared terms “semigraphical translators”

  3. 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 · Shared terms “properly embedded”, “vertical”

  4. 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”

  5. Scherk-like 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”

  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. Notes on translating solitons for Mean Curvature Flow

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

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

  8. Translating solitons of the mean curvature flow asymptotic to hyperplanes in

    Eddygledson S. Gama and Francisco MartinWider flows

    Topics Self-shrinkers & translators · Same author Francisco Martín · Shared terms “grim reaper”, “vertical”

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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.