Paper · Sep 4, 2022

Finite entropy translating solitons in slabs

Eddygledson Souza Gama, Francisco Martín, Niels Martin Møller

math.DGWider flowsarXiv:2209.01640v3
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Abstract

We study translating solitons for the mean curvature flow, which are contained in slabs, and are of finite genus and finite entropy. As a first consequence of our results, we can enumerate connected components of slices to define asymptotic invariants , which count the numbers of "wings". Analyzing these, we give a method for computing the entropies via a simple formula involving the wing numbers, which in particular shows that for this class of solitons the entropy is quantized into integer steps. Finally, combining the concept of wing numbers with Morse theory for minimal surfaces, we prove the uniqueness theorem that if is a complete embedded simply connected translating soliton contained in a slab with entropy and containing a vertical line, then is one of the translating pitchforks of Hoffman-Martín-White

Comments: 40 pages, 3 figures. Final version accepted for publication in the American Journal of Mathematics

Mean curvature flowSelf-shrinkers & translatorsStability & 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 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 Annuloids and Δ-wings (D. Hoffman, F. Martin and B. White, 2023)2Hoffman 2023 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)4Hoffman 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)5Martín 2023 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)3Martín 2024 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)8Chen 2025 Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2025)1Gama 2025 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)6Bueno 2026 The global structure of bi-rotational self-shrinkers (Gregório Silva Neto, 2026)7Neto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 authors Eddygledson Souza Gama, Francisco Martín, Niels Martin Møller · Shared terms “finite entropy”, “finite genus”, “translating solitons”

  2. Annuloids and -wings

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

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

  3. Classification of Semigraphical Translators

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

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

  4. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows PDF

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

  5. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

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

  6. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “translating solitons”, “vertical”

  7. The global structure of bi-rotational self-shrinkers

    Gregório Silva NetoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “finite entropy”

  8. When can minimal hypersurfaces be connected by mean curvature flow?

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “Morse theory”

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) 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.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.