Postprint version. Published in Advances in Mathematics, Volume 280, Issue 6, August 5, 2015, pages 506-548.
The definitive version is available at https://doi.org/10.1016/j.aim.2015.05.001.
Based on [previous publication*], we develop a global correspondence between immersed hypersurfaces ϕ:Mn→Hn+1ϕ:Mn→Hn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e2ρgSne2ρgSn on domains Ω in the boundary SnSn at infinity of Hn+1Hn+1, where ρ is the horospherical support function, ∂∞ϕ(Mn)=∂Ω∂∞ϕ(Mn)=∂Ω, and Ω is the image of the Gauss map G:Mn→SnG:Mn→Sn. To do so we first establish results on when the Gauss map G:Mn→SnG:Mn→Sn is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hn+1Hn+1 and conformal metrics on domains in SnSn. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hn+1Hn+1 of constant mean curvature.
*J.M. Espinar, J.A. Gálvez, P. Mira. Hypersurfaces in Hn+1Hn+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems. J. Eur. Math. Soc. (JEMS), 11 (2009), pp. 903–939