Holomorphic Legendrian Curves in Convex Domains

Abstract

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in C2n+1, n≥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 D⊂C2n+1, may be approximated uniformly on compacts in the interior IntM by holomorphic Legendrian curves IntM→D such that the approximants are proper, complete, agree with the starting curve on a given finite set in IntM 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.