On the volume and the Gauss map image of spacelike hypersurfaces in de Sitter space (Q2781264)

Let \({\mathbf L}^{n+2}\) be the \((n+2)\)-dimensional Lorentz-Minkowski space and \({\mathbf S}_1^{n+1}\subset{\mathbf L}^{n+2}\) be the \((n+1)\)-dimensional unitary de-Sitter space (the standard simply connected Lorentzian space form of positive constant sectional curvature). Let \(\psi:M^n\to{\mathbf S}_1^{n+1}\) be a complete spacelike hypersurface such that its image under the Gauss map is contained in a hyperbolic geodesic ball of radius \(\rho\). Then \(M^n\) is necessarily compact and its \(n\)-dimensional volume satisfies the inequalities \((\ast)\) \(\omega_n/\cosh(\rho)\leq \text{vol}(M)\leq\omega_n\cosh^n(\rho)\), \(\omega_n\) being the volume of a round \(n\)-space of radius one. The possible equalities in \((\ast)\) are characterized. Applications of the above are discussed, among others, the Goddard's conjecture is proved under the assumption that the hyperbolic image of the hypersurface is bounded.