Paper · Jan 22, 2019

Jenkins-Serrin problem for translating horizontal graphs in

Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira, Francisco Martin

math.DGWider flowsarXiv:1901.07224v2
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Abstract

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds . Moreover, we give examples of these graphs in the cases of and .

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Submitted
Jan 22, 2019
Last revised
Jul 29, 2019 (v2)
Subjects
math.DG
DOI
10.4171/rmi/1221

Comments: To appear in Rev. Mat. Iberoam

Mean curvature flowSelf-shrinkers & translators

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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 Translating solitons of the mean curvature flow asymptotic to hyperplanes in ℝⁿ⁺¹ (Eddygledson S. Gama and Francisco Martin, 2018)3Gama 2018 Translating solitons in Riemannian products (Jorge H. de Lira and Francisco Martin, 2018)1de Lira 2018 Notes on translating solitons for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martín et al., 2019)7Hoffman 2019 Translating solitons over Cartan-Hadamard manifolds (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)4Casteras 2020 Non-parametric mean curvature flow with prescribed contact angle in Riemannian products (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)2Casteras 2020 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)5Gama 2022 Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2025)6Gama 2025 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)8de Lira 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating solitons in Riemannian products

    Jorge H. de Lira and Francisco MartinWider flows

    Topics Self-shrinkers & translators · Same authors Jorge H. de Lira, Francisco Martin · Shared terms “Riemannian product”, “translating solitons”

  2. Non-parametric mean curvature flow with prescribed contact angle in Riemannian products

    Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al.Wider flows

    Topics Self-shrinkers & translators · Same authors Esko Heinonen, Jorge H. de Lira · Shared terms “Riemannian product”, “translating solitons”

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

    Eddygledson S. Gama and Francisco MartinWider flows

    Topics Self-shrinkers & translators · Same authors Eddygledson S. Gama, Francisco Martin · Shared terms “translating solitons”

  4. Translating solitons over Cartan-Hadamard manifolds

    Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al.Wider flows

    Topics Self-shrinkers & translators · Same authors Esko Heinonen, Jorge H. de Lira · Shared terms “translating solitons”

  5. Finite entropy translating solitons in slabs

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows

    Topics Self-shrinkers & translators · Same authors Eddygledson S. Gama, Francisco Martin · Shared terms “translating solitons”

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

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows

    Topics Self-shrinkers & translators · Same authors Eddygledson S. Gama, Francisco Martin · Shared terms “translating solitons”

  7. Notes on translating solitons for Mean Curvature Flow

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

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

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

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows

    Topics Self-shrinkers & translators · Same author Jorge H. de Lira · Shared terms “Riemannian product”, “translating solitons”, “product manifolds”

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)

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.