The asymptotic Plateau problem for Hypersurfaces of constant $H_{k}$ curvature in hyperbolic space

In this paper, we study the asymptotic Plateau problem in hyperbolic space for hypersurfaces of constant $H_k$-curvature. We prove the existence of a smooth complete $k$-convex hypersurface in $\mathbb{H}^{n+1}$ satisfying \[ H_k(κ)=σ, \qquad σ\in(0,1), \] with prescribed asymptotic boundary at infi