On stable complete hypersurfaces with vanishing {$r$}-mean curvature

Abstract

A form of Bernstein theorem states that a complete stable minimal surface in euclidean space is a plane. A generalization of this statement is that there exists no complete stable hypersurface of an $n$-euclidean space with vanishing $(n-1)$-mean curvature and nowhere zero Gauss-Kronecker curvature. We show that this is the case, provided the immersion is proper and the total curvature is finite.