Abstract
We consider surfaces of constant Gaussian curvature immersed in \(3\)-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is \(3\)-dimensional hyperbolic space. This allows us to prove results of existence of solutions to the asymptotic Plateau problem, as defined by Labourie, and the continuous dependence of these solutions on the data.
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This paper is a greatly revised version of the second chapter of the author’s doctoral thesis. The author would like to thank François Labourie for having proposed this problem, for his guidance during that period, and for his encouragement to prepare the current version. The author would also like to thank the Université Paris XI, and the Max Planck Institute for Mathematics in the Sciences in Leipzig for providing the conditions required to prepare the previous version of this paper. The author would like thank the Centre de Recerca Matemàtica in Barcelona for providing the conditions required to prepare the current version of this paper which was written whilst the author was benefitting from a Marie Curie Postdoctoral fellowship.
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Smith, G. Hyperbolic Plateau problems. Geom Dedicata 176, 31–44 (2015). https://doi.org/10.1007/s10711-014-9958-2
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DOI: https://doi.org/10.1007/s10711-014-9958-2