Entire spacelike hypersurfaces with constant ${\sigma }_{n-1}$ curvature in Minkowski space

Abstract

We prove that, in 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 ${\mathbb{R}}^{n,1}$ rigid motion, $M$ splits as a product $M^{n-1}\times \mathbb{R}.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant ${\sigma }_{n-1}$ curvature and bounded principal curvatures.