Title

Hypersurfaces in Hyperbolic Space with Support Function

Author Info

Vincent Bonini, California Polytechnic State University - San Luis ObispoFollow
José M. Espinar, Instituto de Matemática Pura e Aplicada
Jie Qint, University of California - Santa Cruz

Recommended Citation

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.

Abstract

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 e^2ρg_Sⁿe2ρgSn 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

Disciplines

Mathematics

Copyright

2015 Elsevier