Paper · Jan 25, 2019

Notes on translating solitons for Mean Curvature Flow

David Hoffman, Tom Ilmanen, Francisco Martín, Brian White

math.DGWider flowsarXiv:1901.09101v4
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Abstract

The purpose of these notes is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in , particularly those which are complete graphs over domains in . In this paper we describe a full classification of complete translating graphs in .

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Submitted
Jan 25, 2019
Last revised
Dec 18, 2021 (v4)
Subjects
math.DG

Comments: Proceedings of the conference M:IV. Version accepted for publication

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 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 Translating solitons of the mean curvature flow asymptotic to hyperplanes in ℝⁿ⁺¹ (Eddygledson S. Gama and Francisco Martin, 2018)7Gama 2018 Graphical Translators for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martin et al., 2018)1Hoffman 2018 Jenkins-Serrin problem for translating horizontal graphs in M ×ℝ (Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al., 2019)8Gama 2019 Nguyen's Tridents and the Classification of Semigraphical Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)4Hoffman 2019 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)6Zhu 2020 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)3Hoffman 2023 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)5Martín 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Graphical Translators for Mean Curvature Flow

    David Hoffman, Tom Ilmanen, Francisco Martin et al.Wider flows

    Topics Self-shrinkers & translators · Same authors David Hoffman, Tom Ilmanen, Francisco Martín · Shared terms “translating graphs”, “complete translating”, “full classification”

  2. Annuloids and -wings

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

    Topics Self-shrinkers & translators · Same authors David Hoffman, Francisco Martín, Brian White · Shared terms “translating graphs”, “complete translating”, “translating solitons”

  3. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

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

  4. Nguyen's Tridents and the Classification of Semigraphical Translators for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

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

  5. Classification of Semigraphical Translators

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

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

  6. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators · Shared terms “translating solitons”

  7. Translating solitons of the mean curvature flow asymptotic to hyperplanes in

    Eddygledson S. Gama and Francisco MartinWider flows

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

  8. Jenkins-Serrin problem for translating horizontal graphs in

    Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al.Wider flows

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

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.

Survey in the Self-shrinkers & translators guide: Hoffman, Ilmanen, Martín and White: notes on translating solitons.

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)

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.