Entire spacelike hypersurfaces with constant $\sigma_k$ curvature in Minkowski space

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.