Let \(\Gamma _{1}\) and \(\Gamma _{2}\) be two disjoint rectifiable star-shaped Jordan curves in the asymptotic boundary \(\partial _{\infty }\mathbb {H}^{3}\) of the hyperbolic space \(\mathbb {H}^3\) . If the distance between \(\Gamma _{1}\) and \(\Gamma _{2}\) is bounded above by the constant given by (1.8), then there exists an area minimizing annulus \(\Pi \subset \mathbb {H}^3\) , which is asymptotic to \(\Gamma _{1}\cup \Gamma _{2}\) . The main results of this paper are Theorem 1.6 and Theorem 1.10.
