  In this work we study spacelike hypersurfaces immersed in spatially open
standard static spacetimes with complete spacelike slices. Under appropriate
lower bounds on the Ricci curvature of the spacetime in directions tangent to
the slices, we prove that every complete CMC hypersurface having either bounded
hyperbolic angle or bounded height is maximal. Our conclusions follow from
general mean curvature estimates for spacelike hypersurfaces. In case where the
spacetime is a Lorentzian product with spatial factor of nonnegative Ricci
curvature and sectional curvatures bounded below, we also show that a complete
maximal hypersurface not intersecting a spacelike slice is itself a slice. This
result is obtained from a gradient estimate for parametric maximal
hypersurfaces.
