On the rigidity of space-like hypersurfaces immersed in the steady state space \(\mathcal H^{n+1}\) (Q2915422)

The authors consider a particular ambient spacetime -- the half \(H^{n+1}\) of the de Sitter space \(S^{n+1}_1\) -- which models the so-called steady state space. Based on the generalized maximum principle of Omori and Yau, they obtain rigidity results concerning complete space-like hypersurfaces immersed in \(S^{n+1}_1\) and study the uniqueness of entire vertical graphs in such ambient spacetimes.NEWLINENEWLINE By imposing a restriction on the normal hyperbolic angle of the hypersurface (that is, the hyperbolic angle between the Gauss map of the hypersurface and the unitary time-like vector field which determines on \(H^{n+1}\) a codimension-one space-like foliation by hyperplanes the authors prove the following:NEWLINENEWLINELet \(\psi\colon \Sigma^n\to H^{n+1} \subset S^{n+1}_1\) be a complete space-like hypersurface bounded away from the future infinite of \(H^{n+1}\), with bounded mean curvature \(H\geq1\). If the normal hyperbolic angle \(\theta\) of \(\Sigma^n\) satisfies \(\cosh\theta\leq\inf_{\Sigma}H\), then \(\Sigma^n\) is a hyperplane and its hyperbolic image is exactly a horosphere.NEWLINENEWLINE They also study the uniqueness of entire vertical graphs in such ambient spacetimes.