Hypersurfaces in ${\mathbb{H}}^{n+1}$ and conformally invariant equations: the generalized Christoffel and Nirenberg problems

Abstract

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in ${\mathbb{H}}^{n_{+}1}$, 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 ${\mathbb{S}}^n$ 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 in 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 ${\mathbb{H}}^{n_{+}1}$ 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 ${\mathbb{S}}^n$ and the theory of hypersurfaces with prescribed hyperbolic Gauss map in ${\mathbb{H}}^{n_{+}1}$. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals $\mathcal{W}({\kappa }_1,\dots ,{\kappa }_n)$, the only compact immersed hypersurfaces in ${\mathbb{H}}^{n_{+}1}$ on which $\mathcal{W}$ is constant are round spheres.