On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space

In this paper we prove that a complete spacelike hypersurface $M^n$ in de Sitter space such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius $\varrho$is necessarily compact and its $n$-dimensional volume satisfies $\omega_n/\mathrm{cosh}(\varrho)\leq \mathrm{vol}(M)\leq\omega_n\mathrm{ cosh}^{n}(\varrho)$, where $\omega_n$ denotes the volume of a unitary round $n$-sphere. We also characterize the case where these inequalities become equalities. As an application of our result, we also conclude that Goddard's conjecture is true under the assumption that the hyperbolic image of the hypersurface is bounded.