Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi–Bernstein-type problems

Abstract

In this article spacelike hypersurfaces immersed in twisted product spacetimes $$I\times _f F$$ I × f F with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, and a relation that needs to hold on each immersed spacelike hypersurface in $$I\times _f F$$ I × f F to be a simple warped product is also derived. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersurfaces and other for totally umbilic ones with a suitable restriction on their mean curvature are presented, and a full description of maximal hypersurfaces in twisted product spacetimes with a one-dimensional Lorentzian fiber is also included. Finally, the mean curvature equation for a spacelike graph on the fiber is computed and as an application, some Calabi–Bernstein-type results are proven.