Paper · Aug 4, 2023

Translating Annuli for Mean Curvature Flow

David Hoffman, Francisco Martín, Brian White

math.DGWider flowsarXiv:2308.02210v4
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Abstract

We construct a family of complete, properly embedded, annular translators such that lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as to four vertical planes and , where . We call and the inner width and the (outer) width of the translator. We show that for each and each , there is a translator in the family with inner width and with necksize . (We also show that there are no translators with inner width having the properties of the examples we construct.)

Comments: 86 pages, 9 figures. To appear in Advances In Mathematics

Mean curvature flowSelf-shrinkers & translators

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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 Uniqueness of the [φ,e₃]-catenary cylinders by their asymptotic behaviour (A. L. Martínez-Triviño and J. P. dos Santos, 2021)7Martínez-Tri 2021 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)4White 2022 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)3Gama 2022 Annuloids and Δ-wings (D. Hoffman, F. Martin and B. White, 2023)1Hoffman 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)6Martín 2023 Rotators-Translators to Mean Curvature Flow in ℍ²×ℝ (Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos Santos, 2024)8de Lima 2024 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)2Martín 2024 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. Annuloids and -wings

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

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

  2. Classification of Semigraphical Translators

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

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

  3. Finite entropy translating solitons in slabs

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

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

  4. 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

  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 · Same author Francisco Martín

  6. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

    Topics Self-shrinkers & translators · Same author Francisco Martín

  7. Uniqueness of the -catenary cylinders by their asymptotic behaviour

    A. L. Martínez-Triviño and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “vertical planes”, “properly embedded”

  8. Rotators-Translators to Mean Curvature Flow in

    Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos SantosWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “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. 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.