Smooth solutions to the Christoffel problem in $\mathbb{H}^{n+1}$

The famous Christoffel problem is possibly the oldest problem of prescribed curvatures for convex hypersurfaces in Euclidean space. Recently, this problem has been naturally formulated in the context of uniformly $h$-convex hypersurfaces in hyperbolic space by Espinar-G\'alvez-Mira. Surprisingly, Espinar-G\'alvez-Mira find that the Christoffel problem in hyperbolic space is essentially equivalent to the Nirenberg-Kazdan-Warner problem on prescribing scalar curvature on $\mathbb{S}^n$. This equivalence opens a new door to study the Nirenberg-Kazdan-Warner problem. In this paper, we establish a existence of solutions to the Christoffel problem in hyperbolic space by proving a full rank theorem. As a corollary, a existence of solutions to the Nirenberg-Kazdan-Warner problem follows.