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Dirichlet problem for f-minimal graphs

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Abstract

We study the asymptotic Dirichlet problem for f-minimal graphs in Cartan-Hadamard manifolds M.f-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 f-minimal graphs with prescribed boundary behavior on a bounded domain Ω ⊂ M under suitable assumptions on f and the boundary of Ω. In the second part, we consider the asymptotic Dirichlet problem. Provided that f decays fast enough, we construct solutions to the problem. Our assumption on the decay of f is linked with the sectional curvatures ofM. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.

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Authors and Affiliations

  1. Departement de Mathématique, Universite Libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050, Bruxelles, Belgium

    Jean-Baptiste Casteras

  2. Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68 (Gustaf Hällströmin Katu 2B), 00014, Helsinki, Finland

    Esko Heinonen & Ilkka Holopainen

Authors
  1. Jean-Baptiste Casteras
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  2. Esko Heinonen
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  3. Ilkka Holopainen
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Corresponding author

Correspondence to Esko Heinonen.

Additional information

J.-B. C. supported by MIS F.4508.14 (FNRS).

E. H. supported by Jenny and Antti Wihuri Foundation.

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Casteras, JB., Heinonen, E. & Holopainen, I. Dirichlet problem for f-minimal graphs. JAMA 138, 917–950 (2019). https://doi.org/10.1007/s11854-019-0051-5

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  • DOI: https://doi.org/10.1007/s11854-019-0051-5

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