Asymptotic Plateau problem for $3$-convex hypersurface in ${\mathbb{H}}^5$

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation ${\prod }_{i=1}^n(H-{\kappa }_i)=((n-1)\sigma {)}^n$ for $n=4$ and has prescribed asymptotic boundary $\mathrm{\Gamma }$ at the infinity of hyperbolic space of dimension 5, where $\sigma \in (0,1)$ is a constant and $\mathrm{\Gamma }$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f(\kappa )=\frac{1}{n-1}({\prod }_{i=1}^n(H-{\kappa }_i){)}^{\frac{1}{n}}$ during uniform global curvature estimate.