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Convexity at infinity in Cartan–Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems

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Abstract

We study the asymptotic Dirichlet and Plateau problems on Cartan–Hadamard manifolds satisfying the so-called strict convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to \(-\infty \) and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to \(-\infty \) faster than any prescribed rate. These improve previous results of Anderson, Borbély, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with \(\mathbb {Z}_2\)-multiplicity in aCartan–Hadamard manifold satisfying the SC condition given any compact topologically embedded \((k-1)\)-dimensional submanifold of \(\partial _{\infty }M,\ 2\le k\le n-1\), as the boundary data. We also solve the asymptotic Plateau problem for locally rectifiable currents with \(\mathbb {Z}\)-multiplicity on any rotationally symmetric manifold satisfying the SC condition given a smoothly embedded submanifold as the boundary data. These generalize previous results of Anderson, Bangert, and Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a large class of PDEs. In particular, we are able to prove the solvability of this problem on manifolds with super-exponential decay (to \(-\infty \)) of the curvature.

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Acknowledgements

The authors would like to thank the International Centre of Theoretical Physics (ICTP), Trieste, Italy, where part of this work has been done, for its support and kind hospitality. We are grateful to Urs Lang for his valuable comments on previous versions of the paper, in particular, for pointing out an incorrect assumption in Theorem  in the first version.

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Authors and Affiliations

  1. Département de Mathématique, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, 1050, Brussels, Belgium

    Jean-Baptiste Casteras

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Ilkka Holopainen

  3. UFRGS, Instituto de Matemática, Av. Bento Goncalves 9500, Porto Alegre, RS, 91540-000, Brazil

    Jaime B. Ripoll

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  1. Jean-Baptiste Casteras
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  2. Ilkka Holopainen
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  3. Jaime B. Ripoll
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Correspondence to Ilkka Holopainen.

Additional information

J.-B.C. supported by the FNRS project MIS F.4508.14; I.H. supported by the Academy of Finland, project 252293; J.R. supported by the CNPq (Brazil) project 302955/2011-9.

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Casteras, JB., Holopainen, I. & Ripoll, J.B. Convexity at infinity in Cartan–Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems. Math. Z. 290, 221–250 (2018). https://doi.org/10.1007/s00209-017-2016-7

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