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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
##### Keywords:
Gauss curvature; immersion; hyperbolic space; embedding
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##### References:
 [1] R. CHARNEY and M. DAVIS, The polar dual of a convex polyhedral set in hyperbolic space, Michigan Math. J., 42 (1995), 479-510. · Zbl 0860.52007 [2] F. LABOURIE, Immersions isométriques elliptiques et courbes pseudo-holomorphes, Journal of Differential Geometry, 30 (1989), 395-424. · Zbl 0682.53063 [3] F. LABOURIE and J.-M. SCHLENKER, Surfaces convexes fuchsiennes dans LES espaces lorentziens à courbure constante, Prépublication 96-05, Université de Paris-Sud, 1996. · Zbl 0864.53016 [4] A. V. POGORELOV, Extrinsic geometry of convex surfaces. American Mathematical Society, 1973, Translations of Mathematical Monographs, Vol. 35. · Zbl 0311.53067 [5] I. RIVIN and C. D. HODGSONA characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math., 111 (1993), 77-111. · Zbl 0784.52013 [6] H. ROSENBERG and J. SPRUCK, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, Journal of Differential Geometry, 40 (1994), 379-409. · Zbl 0823.53047 [7] J.-M. SCHLENKER, Métriques sur LES polyèdres hyperboliques convexes, Prépublication no 97-18, Université de Paris-Sud, à paraître, Journal of Differential Geometry. · Zbl 0912.52008 [8] J.-M. SCHLENKER, Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. in Analysis and Geometry, 4 (1996), 285-331. · Zbl 0864.53016 [9] M. SPIVAK, A comprehensive introduction to geometry, Vol. I-V, publish or perish, 1970-1975. · Zbl 0306.53001
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