The famous Christoffel problem is possibly the oldest problem of prescribed curvatures for convex hypersurfaces in Euclidean space. Recently, Espinar-Galvez-Mira have formulated this problem in the context of uniformly h-convex hypersurfaces in hyperbolic space.
Surprisingly, Espinar-Galvez-Mira found that the Christoffel problem in hyperbolic space is essentially equivalent to the Nirenberg problem on prescribed scalar curvature on the unit sphere. This equivalence provides a new approach to the Nirenberg problem.
In this talk, we will establish a existence of solutions to the Christoffel problem in hyperbolic space by proving a full rank theorem . As a corollary, a new existence result for the Nirenberg problem is obtained.
