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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.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57R40 Embeddings in differential topology
Full Text: DOI Numdam EuDML
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