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Hypersurfaces in Hyperbolic Space with Support Function
Advances in Mathematics
  • Vincent Bonini, California Polytechnic State University - San Luis Obispo
  • José M. Espinar, Instituto de Matemática Pura e Aplicada
  • Jie Qint, University of California - Santa Cruz
Publication Date
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
Citation Information
Vincent Bonini, José M. Espinar and Jie Qint. "Hypersurfaces in Hyperbolic Space with Support Function" Advances in Mathematics Vol. 280 Iss. 6 (2015) p. 506 - 548
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