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

TL;DR

This paper proves the existence of solutions to the Christoffel problem in hyperbolic space, revealing a deep connection to the Nirenberg-Kazdan-Warner problem and advancing understanding of prescribed curvature problems.

Contribution

It establishes a full rank theorem for the Christoffel problem in hyperbolic space, leading to new existence results and linking to the Nirenberg-Kazdan-Warner problem.

Findings

Existence of solutions to the Christoffel problem in hyperbolic space.

Full rank theorem for the problem.

Corollary: existence of solutions to the Nirenberg-Kazdan-Warner problem.

Abstract

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.