Holomorphic Legendrian Curves in Convex Domains

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in \(\mathbb {C}^{2n+1}\) ${\mathbb{C}}^{2n+1}$ , \(n \ge 2\) $n\ge 2$ , 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 \(\mathscr {D}\subset \mathbb {C}^{2n+1}\) $\mathcal{D}\subset {\mathbb{C}}^{2n+1}$ , may be approximated uniformly on compacts in the interior \({{\,\textrm{Int}\,}}M\) $\text{Int}M$ by holomorphic Legendrian curves \({{\,\textrm{Int}\,}}M \rightarrow \mathscr {D}\) $\text{Int}M\to \mathcal{D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in \({{\,\textrm{Int}\,}}M\) $\text{Int}M$ 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 compact bordered Riemann surface properly embeds into a convex domain as a complete curve under a suitable geometric condition on the boundary of the codomain.