On the geometry of horospheres

Abstract

The aim of this paper is to investigate Bernstein-type properties of horospheres of the hyperbolic space ${\mathbb{H}}^{n+1}$. Our approach is based on the use of appropriate generalized maximum principles in order to obtain new characterization results of such horospheres. Furthermore, by supposing a linear dependence between support functions naturally attached to a hypersurface, we also establish a classification theorem concerning horospheres and hyperbolic cylinders of ${\mathbb{H}}^{n+1}$.