Holomorphic Legendrian curves in convex domainsSvetina, Andrej (Avtor) holomorphic Legendrian curveconvex domaincomplete Legendrian embeddingWe prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in ${\mathbb C}^{2n+1}$, $n \ge 2$, 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 ${\mathcal D} \subset {\mathbb C}^{2n+1}$, may be approximated uniformly on compacts in the interior ${\rm Int} M$ by holomorphic Legendrian curves ${\rm Int} M \to {\mathcal D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in ${\rm Int} M$ 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.20252025-10-17 14:40:17Članek v reviji175141sl