Hypersurfaces with prescribed curvatures in the de Sitter space

Abstract

In this paper, we establish a relationship between spacelike hypersurfaces in the de Sitter space ${\mathbb{S}}_1^{n+1}$ and conformal metrics on the sphere ${\mathbb{S}}^n$. As a consequence of this relation and some deep results in conformal geometry, we classify spacelike hypersurfaces in ${\mathbb{S}}_1^{n+1}$ satisfying certain global conditions and a very general Weingarten relation $W(k_1,\dots ,k_n)=0$ of elliptic type, where $k_1,\dots ,k_n$ are its principal curvatures. Also, we particularize to some important cases as the $r$th mean curvatures.