On the asymptotic Plateau problem for area minimizing surfaces in $\mathbb{E}(-1,\tau )$

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,\tau )$. As one of our main results, we present sufficient conditions for a curve $\mathrm{\Gamma }$ in ${\mathrm{\partial }}_{\mathrm{\infty }}\mathbb{E}(-1,\tau )$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,\tau )$ having $\mathrm{\Gamma }$ as its asymptotic boundary.