Hypersurfaces in Hn and the space of its horospheres
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A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S2 with curvature K > —1 is induced on a unique convex surface in H3. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric.¶This result extends in higher dimensions, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced metric or the third fundamental form become fairly easy when one considers the horospherical metric, which thus provides a good boundary condition for the construction of hyperbolic metrics on a manifold with boundary.¶The results concerning the third fundamental form are obtained using a duality between H3 and the de Sitter space \( S^3_1 \). In the same way, the results concerning the horospherical metric are proved through a duality between Hn and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure.
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