Abstract.
A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S 2 with curvature K > —1 is induced on a unique convex surface in H 3. 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 H 3 and the de Sitter space \( S^3_1 \). In the same way, the results concerning the horospherical metric are proved through a duality between H n and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Additional information
Submitted: March 2001, Revised: November 2001.
Rights and permissions
About this article
Cite this article
Schlenker, JM. Hypersurfaces in Hn and the space of its horospheres . GAFA, Geom. funct. anal. 12, 395–435 (2002). https://doi.org/10.1007/s00039-002-8252-x
Issue Date:
DOI: https://doi.org/10.1007/s00039-002-8252-x