THE CHRISTOFFEL PROBLEM IN LORENTZIAN GEOMETRY

Abstract

The Christoffel problem, in its classical formulation, asks for a characterization of real functions defined on the unit sphere $S^{n-1}\subset\mathbb{R}^n$ which occur as the mean curvature radius, expressed in terms of the Gauss unit normal, of a closed convex hypersurface, i.e. the boundary of a convex body in $\mathbb{R}^n$. In this work we consider the related problem in Lorentz space $\mathbb{L}^n$ and present necessary and sufficient conditions for a $C^1$ function defined in the hyperbolic space $H^{n-1}\subset\mathbb{L}^n$ to be the mean curvature radius of a spacelike embedding $\bm{M}\hookrightarrow\mathbb{L}^n$.