Hyperbolic complex contact structures on ${\mathbb{C}}^{2n+1}$

In this paper we construct complex contact structures on ${\mathbb{C}}^{2n+1}$ for any $n\ge 1$ with the property that every holomorphic Legendrian map $\mathbb{C}\to {\mathbb{C}}^{2n+1}$ is constant. In particular, these contact structures are not globally contactomorphic to the standard complex contact structure on ${\mathbb{C}}^{2n+1}$.