Paper · Sep 14, 2025

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

Eddygledson Souza Gama, Francisco Martín, Niels Martin Møller

math.DGWider flowsarXiv:2509.11473v3
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Abstract

We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time , any finite entropy, finite genus, embedded, collapsed translating soliton in converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with -data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as .

Comments: 54 pages, 10 figures; further citations added, exposition improved

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & non-compact

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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)6Martínez-Tri 2021 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)8White 2022 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)1Gama 2022 Annuloids and Δ-wings (D. Hoffman, F. Martin and B. White, 2023)5Hoffman 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)4Martín 2023 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)2Martín 2024 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)7Daniels-Holg 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite entropy translating solitons in slabs

    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. 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

  3. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows PDF

    Topics Self-shrinkers & translators · Same author Francisco Martín

  4. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

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

  5. Annuloids and -wings

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

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

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

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

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “grim reaper”, “translating solitons”, “elliptic”

  7. Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

    J. M. Daniels-Holgate and Or HershkovitsWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact, Singularities

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

    Brian WhiteWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “Huisken”

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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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, 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)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.