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

Abstract

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.

Final version, accepted for publication on Ann. Global Anal. Geom. 19 pages, 6 figures