Based on [previous publication*], we develop a global correspondence between immersed hypersurfaces ϕ:M^{n}→H^{n+1}ϕ:Mn→Hn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e^{2ρ}g_{Sn}e2ρgSn on domains Ω in the boundary S^{n}Sn at infinity of H^{n+1}Hn+1, where *ρ * is the horospherical support function, ∂_{∞}ϕ(M^{n})=∂Ω∂∞ϕ(Mn)=∂Ω, and Ω is the image of the Gauss map G:M^{n}→S^{n}G:Mn→Sn. To do so we first establish results on when the Gauss map G:M^{n}→S^{n}G: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 H^{n+1}Hn+1 and conformal metrics on domains in S^{n}Sn. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in H^{n+1}Hn+1 of constant mean curvature.
*J.M. Espinar, J.A. Gálvez, P. Mira. Hypersurfaces in H^{n+1}Hn+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems. J. Eur. Math. Soc. (JEMS), 11 (2009), pp. 903–939

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