2007/06/15 by José M. Espinar, J. M. Espinar, J. A. Galvez +6 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.0706.2314
37 pages
arxiv created 2007/06/15 · openalex publication_date 2007/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in Hn+1, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on §n for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result into a more general framework. Specifically, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in Hn+1 is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on §n and the theory of hypersurfaces with prescribed hyperbolic Gauss map in \Hn+1. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals W(\k1,..., \kn), the only compact immersed hypersurfaces in Hn+1 on which W is constant are round spheres.