Umbilicity of complete linear Weingarten hypersurfaces immersed in certain Einstein manifolds

Abstract

In this paper, we present some sets of geometric conditions that ensure the umbilicity of linear Weingarten hypersurfaces immersed in an Einstein manifold satisfying standard curvature constraints which, in particular, are verified by a Riemannian space with constant sectional curvature. Our methodology relies as an application of maximum principles involving concepts of convergence to zero at infinity and polynomial volume growth, both of them related to complete noncompact Riemannian manifolds.