Estimates for the curvatures of spacelike hypersurfaces in gradient conformally stationary spacetimes

Abstract

In this paper we establish some estimates for the curvatures of complete spacelike hypersurfaces in a gradient conformally stationary (GCS) spacetime which are contained in certain unbounded regions of determined by suitable physical assumptions. By curvatures we mean the higher order mean curvatures of the hypersurface, as well as its scalar and Ricci curvatures. Our results will be a consequence of the generalized maximum principle. In this sense, we provide new natural geometrical conditions to allow one to apply the generalized maximum principle to complete spacelike hypersurfaces in generalized Robertson–Walker spacetimes, which constitute an important subclass of GCS spacetimes.