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Réalisations de surfaces hyperboliques complètes dans \(H^3\). (Realizations of complete hyperbolic surfaces in \(H^3\)). (French) Zbl 0916.53029
The author treats complete metrics of Gaussian curvature \(K_0\) with \(K_0\in\;]-1,0[,\) defined on a surface \(\Sigma\) which is diffeomorphic to the sphere \(S^2\) minus \(N\) points, \(N\geqslant 3\). He considers the question of embedding into the hyperbolic space \(H^3\) and he demonstrates the following.
Given a complete such metric, say \(\sigma,\) with each end having infinite area, then there exists an embedding \(\phi :\Sigma\rightarrow H^3\) with induced metric \(\sigma\), whose asymptotic boundary is the union of disjoint circles of \(\partial_\infty H^3\). Furthermore, \(\phi\) is unique modulo rigid motions of \(H^3\).
There is a more precise result (and other related theorems) that the interested reader can find in the paper.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57R40 Embeddings in differential topology
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