In 1982, Treibergs showed that every entire spacelike CMC hypersurface is convex and has bounded principal curvatures. It is natural to ask, do constant entire spacelike $\sigma_k$ curvature hypersurfaces share similar properties as CMC hypersurfaces?
In this talk, we show that, in the Minkowski space, if a spacelike, (n − 1)-convex hypersurface $M$ with constant $\sigma_{n−1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an $R^ {n,1}$ rigid motion, $M$ splits as a product $M^{n−1}\times R$. We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant $\sigma_{n−1}$ curvature and bounded principal curvatures. This is a joint work with Changyu Ren and Zhizhang Wang.