Entire spacelike hypersurfaces with constant \(\sigma_k\) curvature in Minkowski space (Q2122067)

Minkowski spaces and their time-like/space-like CMC (constant mean curvature) and constant Gaussian curvature surfaces have been widely investigated by many mathematicians. More specifically, the papers [\textit{A. E. Treibergs}, Invent. Math. 66, 39--56 (1982; Zbl 0483.53055); \textit{H. I. Choi} and \textit{A. Treibergs}, J. Differ. Geom. 32, No. 3, 775--817 (1990; Zbl 0717.53038)] can be considered as the starting points of the subject investigated in this paper. In this interesting paper, the authors study the existence of smooth, entire, strictly convex, space-like, constant \(\sigma_k\) curvature hypersurfaces with prescribed light-like directions. Furthermore, they generalize the work done in [\textit{C. Ren} et al., ``Entire space-like hypersufaces with constant \(\sigma_{n-1}\) curvature in Minkowski space'', Preprint, \url{arXiv:2005.06109}] and construct strictly convex, space-like, constant \(\sigma_k\) curvature hypersurfaces with bounded principal curvature, whose image of the Gauss map is the unit ball. The main theorems of the paper are: Theorem. Suppose that \(\mathcal{F} \subset \mathbb{S}^{n-1}\) is the closure of an open subset and \(\partial \mathcal{F} \in C^{1,1}\). Then for \(1 < k < n\), there exists a smooth, entire, space-like, strictly convex hypersurface \(\mathcal{M}_u = \{(x, u(x))| x \in \mathbb{R}^n\}\) satisfying \[ \sigma_k(\kappa[\mathcal{M}_u])=\binom{n}{k}, \tag{1} \] where \(\kappa[\mathcal{M}_u] = (\kappa_1, \kappa_2, \dots, \kappa_n)\) is the principal curvatures of \(\mathcal{M}_u\). Moreover, when \(\frac{x}{|x|}\in \mathcal{F}\), there holds \[ u(x) \rightarrow |x|, \text{ as } |x| \rightarrow \infty. \] Further, the Gauss map image of \(\mathcal{M}_u\) is the convex hull \(\mathrm{Conv}(\mathcal{F})\) of \(\mathcal{F}\) in the unit disc. Theorem. Given any \(f \in C^2(\mathbb{S}^{n-1})\), there is a unique, space-like, strictly convex hypersurface \(\mathcal{M}_u = \{(x, u(x))| x \in \mathbb{R}^n\} \) with bounded principle curvatures satisfying Equation (1). Moreover, \[ u(x) \rightarrow |x|+f\Big( \frac{x}{|x|} \Big), \text{ as }|x| \rightarrow \infty. \] Furthermore, the Gauss map image \(\mathcal{M}_u\) is the open unit disc.