The exterior Dirichlet Problem for homogeneous complex $k$-Hessian equation

In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain ${\mathbb{C}}^n\setminus \mathrm{\Omega }$. We prove the existence and uniqueness of the $C^{1,1}$ solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second order estimate.