Paper · Dec 28, 2019

Simply connected translating solitons contained in slabs

Francesco Chini

math.DGWider flowsarXiv:1912.12426v2
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Abstract

In this work we show that -dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of with entropy strictly less than must be mean convex and thus, thanks to a result by J. Spruck and L. Xiao, are convex. Recently, such -dimensional convex translating solitons have been completely classified by Hoffman, Ilmanen, Martín and White, up to an ambient isometry, as vertical plane, (tilted) grim reaper cylinders, -wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples show that the bound on the entropy is necessary.

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Submitted
Dec 28, 2019
Last revised
Jan 21, 2020 (v2)
Subjects
math.DG, math.AP

Comments: 25 pages, 2 figures, comments are welcome

Mean curvature flowSelf-shrinkers & translatorsRotational symmetry & warped products

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 Some results on translating solitons of the mean curvature flow (Jesús Pérez-García, 2016)4Pérez-García 2016 Bi-Halfspace and Convex Hull Theorems for Translating Solitons (Francesco Chini and Niels Martin Møller, 2018)6Chini 2018 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)2Zhu 2021 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)1Gama 2022 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)8Raji 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)7de Lima 2026 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)5de Lira 2026 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)3Davila 2026
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

    Topics Self-shrinkers & translators · Shared terms “connected translating”, “soliton contained”, “translating solitons”

  2. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “convex translating”, “translating solitons”, “mean convex”

  3. A Bernstein problem for translating solutions to the mean curvature flow

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “convex translating”, “symmetric bowl”, “translating solitons”

  4. Some results on translating solitons of the mean curvature flow

    Jesús Pérez-GarcíaWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”

  5. Translating solitons to higher order mean curvature flows in Riemannian products

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”, “rotationally symmetric”

  6. Bi-Halfspace and Convex Hull Theorems for Translating Solitons

    Francesco Chini and Niels Martin MøllerWider flows

    Topics Self-shrinkers & translators · Same author Francesco Chini · Shared terms “translating solitons”, “Hoffman”, “vertical”

  7. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”, “White”

  8. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “rotationally symmetric”

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.