The Convexity of Entire Spacelike Hypersurfaces with Constant \(\sigma _{n-1}\) Curvature in Minkowski Space

We prove that, in Minkowski space, if a spacelike, \((n-1)\) $(n-1)$ -convex hypersurface \(\mathcal {M}\) $\mathcal{M}$ with constant \(\sigma _{n-1}\) ${\sigma }_{n-1}$ curvature has bounded principal curvatures, then \(\mathcal {M}\) $\mathcal{M}$ is convex. Moreover, if \(\mathcal {M}\) $\mathcal{M}$ is not strictly convex, after an \(\mathbb {R}^{n,1}\) ${\mathbb{R}}^{n,1}$ rigid motion, \(\mathcal {M}\) $\mathcal{M}$ splits as a product \(\mathcal {M}^{n-1}\times \mathbb {R}.\) ${\mathcal{M}}^{n-1}\times \mathbb{R}.$