Weak maximum principles and geometric estimates for spacelike hypersurfaces in generalized Robertson–Walker spacetimes

In this paper we deal with complete spacelike hypersurfaces in a generalized Robertson–Walker spacetime. Using as main analytical tool a new local form of the weak maximum principle for a class of operators including the Lorentzian mean curvature operator, we obtain some mean curvature estimates and height estimates for spacelike graphs with nice Bernstein type consequences. We also give height estimates for spacelike hypersurfaces with constant higher order mean curvature, which are obtained via the local form of the weak maximum principle recently given in Alías et al. (in press) for a class of operators including those constructed from the Newton tensors of a hypersurface.