Paper · Nov 25, 2024

Classification of Semigraphical Translators

Francisco Martín, Mariel Sáez, Raphael Tsiamis, Brian White

math.DGWider flowsarXiv:2411.16889v2
PDF Abstract page

As on arXiv

Abstract

We complete the classification of semigraphical translators for mean curvature flow in that was initiated by Hoffman-Martín-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Radó theory for translators and an angular maximum principle.

Comments: 13 pages, 2 figures. This paper supersedes the preprint arXiv:2310.06980. Final version to appear in Crelle

Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness

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 2021, 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 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)6White 2022 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)4Gama 2022 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)8de Lima 2022 Annuloids and Δ-wings (D. Hoffman, F. Martin and B. White, 2023)3Hoffman 2023 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)7de Lima 2023 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)2Hoffman 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)1Martín 2023 Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2025)5Gama 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Same authors Francisco Martín, Mariel Sáez, Raphael Tsiamis · Shared terms “semigraphical translators”, “helicoid translators”, “Morse Radó”

  2. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows PDF

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

  3. Annuloids and -wings

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

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

  4. Finite entropy translating solitons in slabs

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows PDF

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

  5. Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Francisco Martín

  6. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows PDF

    Topics Self-shrinkers & translators · Same author Brian White

  7. Solitons to Mean Curvature Flow in the hyperbolic 3-space

    R. F. de Lima, A. K. Ramos and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “Specifically”

  8. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness

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. 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) 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)

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.