Integral formulas for compact space-like hypersurfaces in de Sitter space: Applications to the case of constant higher order mean curvature

Abstract

In this paper we develop some integral formulas for compact space-like hypersurface in de Sitter space S 1 n+1 and apply them in order to characterized thetotally umbilical round spheres of S 1 n+1 as the only compact space-like hypersurfaces with constant higher order mean curvature under some appropriate hypothesis. In particular, for hypersurfaces contained in the chronological future (or past) of an equator of S 1 n+1 we prove that the only compact space-like hypersurface with a constant higher order mean curvature are the totally umbilical round spheres.