Curvature estimates for semi-convex solutions of asymptotic Plateau problem in ${\mathbb{H}}^{n+1}$

In this paper, we consider the asymptotic ${\sigma }_k$ Plateau problem in hyperbolic space. We establish $C^2$ estimates for semi-convex complete hypersurfaces satisfying constant ${\sigma }_k$ curvature with a prescribed asymptotic boundary at the infinity for $2\le k\le n-2$ . The result is based on a new crucial concavity inequality derived for hessian equations.