On the geometry of conformally stationary Lorentz spaces

We study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending a result of J. Simons on the minimality of cones in Euclidean space to these spaces, and apply it to the construction of complete, noncompact minimal Lorentz submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong r -stability of spacelike hypersurfaces of constant r -th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.