Paper · May 15, 2020

Ruled surfaces of generalized self-similar solutions of the mean curvature flow

Rafael López

math.DGWider flowsarXiv:2005.07660
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Abstract

In Euclidean space, we investigate surfaces whose mean curvature satisfies the equation , where is the Gauss map, is the position vector and and are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.

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Submitted
May 15, 2020
Subjects
math.DG
Mean curvature flowSelf-shrinkers & translatorsSurfaces

Its neighbourhood

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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 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 Complete Lagrangian self-shrinkers in mathbf R⁴ (Qing-Ming Cheng, Hiroaki Hori and Guoxin Wei, 2018)8Cheng 2018 Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2018)7Bernstein 2018 A family of MCF solutions for the Heisenberg Group (Benedito Leandro, Adriana Araujo Cintra and Hiuri Fellipe dos Santos Reis, 2019)4Leandro 2019 A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space (Erdem Kocakuşaklı and Miguel Ortega, 2020)5Kocakuşaklı 2020 Stability properties of complete self-shrinking surfaces in ℝ³ (Hilário Alencar, Gregório Silva Neto and Detang Zhou, 2021)6Alencar 2021 Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space (Gregório Silva Neto and Vanessa Silva, 2023)1Neto 2023 Invariant λ-translators for the Gauss curvature flow in Euclidean space (Muhittin Evren Aydin and Rafael López, 2025)2Aydin 2025 Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in ℍ³ as Translation Surfaces (Tarcios Andrey Ferreira and João Paulo dos Santos, 2025)3Ferreira 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

    Gregório Silva Neto and Vanessa SilvaWider flows

    Topics Self-shrinkers & translators · Shared terms “ruled surfaces”, “Euclidean”

  2. Invariant -translators for the Gauss curvature flow in Euclidean space

    Muhittin Evren Aydin and Rafael LópezWider flows

    Topics Self-shrinkers & translators · Same author Rafael López · Shared terms “Gauss map”, “Euclidean space”, “translations”

  3. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

    Tarcios Andrey Ferreira and João Paulo dos SantosWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “translation surfaces”

  4. A family of MCF solutions for the Heisenberg Group

    Benedito Leandro, Adriana Araujo Cintra and Hiuri Fellipe dos Santos ReisWider flows

    Topics Self-shrinkers & translators · Shared terms “ruled surfaces”

  5. A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space

    Erdem Kocakuşaklı and Miguel OrtegaWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “translation surfaces”

  6. Stability properties of complete self-shrinking surfaces in

    Hilário Alencar, Gregório Silva Neto and Detang ZhouWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “Euclidean space”

  7. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “self expanders”

  8. Complete Lagrangian self-shrinkers in

    Qing-Ming Cheng, Hiroaki Hori and Guoxin WeiWider flows

    Topics Surfaces, Self-shrinkers & translators · Shared terms “Euclidean space”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.