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 com- plete 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 immer- sion is proper and the total curvature is finite.