Entire spacelike hypersurfaces with constant σk 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 Ren et al. (Entire spacelike hypersufaces with constant sigma(n-1) curvature in Minkowski space. Preprint, arXiv:2005.06109) and construct strictly convex, spacelike, constant sigma(k) curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.