Paper · Jul 6, 2020

Translating solitons over Cartan-Hadamard manifolds

Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen, Jorge H. de Lira

math.DGWider flowsarXiv:2007.02989v2
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Abstract

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.

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Submitted
Jul 6, 2020
Last revised
Jul 12, 2021 (v2)
Subjects
math.DG
Journal
The Journal of Geometric Analysis 33, 163 (2023)
DOI
10.1007/s12220-023-01218-9

Comments: This replaces the first version. We have deleted the whole Section 3 from the previous version due to a gap in a proof. We are grateful to Dr. Hengyu Zhou for pointing out the gap in the proof of Lemma 3.3 in the previous version

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 Dirichlet problem for f-minimal graphs (Jean-Baptiste Casteras, Esko Heinonen and Ilkka Holopainen, 2016)1Casteras 2016 Translating solitons in Riemannian products (Jorge H. de Lira and Francisco Martin, 2018)8de Lira 2018 Jenkins-Serrin problem for translating horizontal graphs in M ×ℝ (Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al., 2019)4Gama 2019 Non-parametric mean curvature flow with prescribed contact angle in Riemannian products (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)5Casteras 2020 Minkowski inequality in Cartan-Hadamard manifolds (Mohammad Ghomi and Joel Spruck, 2022)7Ghomi 2022 Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds (Jacob Bernstein and Arunima Bhattacharya, 2022)6Bernstein 2022 Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds (Claudia Fernandes, Jorge de Lira and Matheus Soares, 2026)2Fernandes 2026 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)3de Lira 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Dirichlet problem for -minimal graphs

    Jean-Baptiste Casteras, Esko Heinonen and Ilkka HolopainenWider flows

    Topics Self-shrinkers & translators · Same authors Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen · Shared terms “asymptotic Dirichlet”, “Hadamard manifolds”, “Cartan Hadamard”

  2. Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds

    Claudia Fernandes, Jorge de Lira and Matheus SoaresWider flows

    Topics Mean curvature flow · Same author Jorge H. de Lira · Shared terms “Hadamard manifolds”, “Cartan Hadamard”, “asymptotic Dirichlet”

  3. 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 “Hadamard manifolds”, “translating solitons”, “Cartan Hadamard”

  4. 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 authors Esko Heinonen, Jorge H. de Lira · Shared terms “translating solitons”

  5. 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 Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen · Shared terms “translating solitons”, “graphical”

  6. Colding-Minicozzi Entropies in Cartan-Hadamard Manifolds

    Jacob Bernstein and Arunima BhattacharyaWider flows

    Topics Mean curvature flow · Shared terms “Hadamard manifolds”, “Cartan Hadamard”

  7. Minkowski inequality in Cartan-Hadamard manifolds

    Mohammad Ghomi and Joel SpruckWider flows

    Topics Mean curvature flow · Shared terms “Cartan Hadamard”, “Hadamard manifolds”

  8. Translating solitons in Riemannian products

    Jorge H. de Lira and Francisco MartinWider flows

    Topics Self-shrinkers & translators · Same author Jorge H. de Lira · 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.

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.