Entire spacelike constant $\sigma_k$ curvature hypersurfaces with prescribed boundary data at infinity

In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $\sigma_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $\sigma_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\mathcal{F},$ as $|x|\rightarrow\infty,$ $u(x)-|x|\rightarrow q\left(\frac{x}{|x|}\right).$ This result is new even for the case of constant Gauss curvature. We also prove that when the Gauss map image is a half disc $\bar{B}_1^+$ and the perturbation $q\equiv 0,$ if a CMC hypersurface $\mathcal{M}_u$ satisfies $|u(x)-V_{\bar{\mathcal{B}}_+}(x)|$ is bounded, then $u(x)$ is unique.