Holomorphic Legendrian curves in convex domains

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in $\mathbb{C}^{2n+1}$, $n \geq 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface $M$, whose image lies in the interior of a convex domain $\mathscr{D} \subset \mathbb{C}^{2n+1}$, may be approximated uniformly on compacts in the interior $\mathrm{Int} \, M$ by holomorphic Legendrian curves $\mathrm{Int} \, M \to \mathscr{D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in $\mathrm{Int} \, M$ to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any bordered Riemann surface properly embeds into a convex domain as a complete holomorphic Legendrian curve under a suitable geometric condition on the boundary of the codomain.