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 with arbitrary complex structure in the real hyperbolic space \(\mathbb {H}^3\).