  In this paper, we prove the existence of smooth, entire, strictly convex,
spacelike, constant $\sigma_k$ curvature hypersurfaces with prescribed
lightlike directions in Minkowski space. This is equivalent to prove the
existence of smooth, entire, strictly convex, spacelike, constant $\sigma_k$
curvature hypersurfaces with prescribed Gauss map image. We also show that
there doesn't exist any entire, convex, strictly spacelike, constant $\sigma_k$
curvature hypersurfaces. Moreover, we generalize the result in \cite{RWX} and
construct strictly convex, spacelike, constant $\sigma_k$ curvature
hypersurface with bounded principal curvature, whose image of the Gauss map is
the unit ball.
