Paper · May 6, 2016

Dirichlet problem for -minimal graphs

Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen

math.DGWider flowsarXiv:1605.01935v2
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Abstract

We study the asymptotic Dirichlet problem for -minimal graphs in Cartan-Hadamard manifolds . -minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of -minimal graphs with prescribed boundary behavior on a bounded domain under suitable assumptions on and the boundary of . In the second part, we consider the asymptotic Dirichlet problem. Provided that decays fast enough, we construct solutions to the problem. Our assumption on the decay of is linked with the sectional curvatures of . In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.

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Submitted
May 6, 2016
Last revised
Jul 25, 2019 (v2)
Subjects
math.DG
DOI
10.1007/s11854-019-0051-5

Comments: Final version, to appear in Journal d'Analyse Mathématique

Mean curvature flowSelf-shrinkers & translatorsMinimal & CMC surfaces

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 Existence and non-existence of minimal graphs (Qi Ding, J. Jost and Y. L. Xin, 2017)2Ding 2017 The global geometry of surfaces with prescribed mean curvature in ℝ³ (Antonio Bueno, Jose A. Galvez and Pablo Mira, 2018)7Bueno 2018 Index and first Betti number of f-minimal hypersurfaces and self-shrinkers (Debora Impera, Michele Rimoldi and Alessandro Savo, 2018)5Impera 2018 Translating solitons over Cartan-Hadamard manifolds (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)1Casteras 2020 Non-parametric mean curvature flow with prescribed contact angle in Riemannian products (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)3Casteras 2020 The Dirichlet problem for the minimal surface system on smooth domains (Caiyan Li and Hengyu Zhou, 2025)4Li 2025 Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds (Claudia Fernandes, Jorge de Lira and Matheus Soares, 2026)6Fernandes 2026 The global structure of bi-rotational self-shrinkers (Gregório Silva Neto, 2026)8Neto 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 over Cartan-Hadamard manifolds

    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 “asymptotic Dirichlet”, “Hadamard manifolds”, “Cartan Hadamard”

  2. Existence and non-existence of minimal graphs

    Qi Ding, J. Jost and Y. L. XinWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal graphs”, “prescribed boundary”, “Dirichlet”

  3. 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

  4. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “Dirichlet”, “bounded domain”

  5. Index and first Betti number of -minimal hypersurfaces and self-shrinkers

    Debora Impera, Michele Rimoldi and Alessandro SavoWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “minimal hypersurfaces”

  6. 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 · Shared terms “asymptotic Dirichlet”, “Hadamard manifolds”, “Cartan Hadamard”

  7. The global geometry of surfaces with prescribed mean curvature in

    Antonio Bueno, Jose A. Galvez and Pablo MiraWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

  8. The global structure of bi-rotational self-shrinkers

    Gregório Silva NetoWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.