Upper and Lower Bounds for the Volume of a Compact Spacelike Hypersurface in a Generalized Robertson–walker Spacetime

We provide upper and lower bounds for the volume of a compact spacelike hypersurface in an (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime in terms of the volume of the fiber, the hyperbolic angle function and the warping function. Under several geometrical and physical assumptions, we characterize the spacelike slices as the only spacelike hypersurfaces where these bounds are attained. As a consequence of these results, we get an upper bound for the first eigenvalue of a compact spacelike surface in a three-dimensional GRW spacetime whose fiber is a topological sphere, which includes the case of the three-dimensional De Sitter spacetime, and show that the bound is attained if and only if M is a spacelike slice.