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 $\Gamma$ in $\partial_{\infty} \mathbb{E}(-1,\tau)$ 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 $\Gamma$ as its asymptotic boundary.