Asymptotic Plateau problem for $2$-convex hypersurface in $\mathbb{H}^4$

Abstract: We prove the existence of smooth complete $2$-convex hypersurface which satisfies prescribed curvature equation $(\kappa_1 + \kappa_2)(\kappa_1 + \kappa_3)(\kappa_2 + \kappa_3) = (2 \sigma)^3$ and has prescribed asymptotic boundary at infinity of hyperbolic space of dimension 4, where $\sigma \in (0, 1)$ is a constant. We also prove the existence for $\sigma_k (\kappa_2 + \cdots + \kappa_n, \cdots, \kappa_1 + \cdots + \kappa_{n - 1}) = C_n^k (n - 1)^k \sigma^k$ with $k < n$ in $\mathbb{H}^{n + 1}$.