Proper holomorphic Legendrian curves in $SL_2(\mathbb{C})$

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts in the real hyperbolic space $\mathbb{H}^3$ with arbitrary complex structure.