Holomorphic Legendrian curves in convex domains

TL;DR

This paper establishes approximation and interpolation results for holomorphic Legendrian curves in convex domains in complex Euclidean spaces, enabling their proper embedding and completeness with prescribed boundary behavior.

Contribution

It introduces new approximation and interpolation techniques for Legendrian curves in convex domains, extending to arbitrary convex domains and ensuring proper, complete embeddings.

Findings

Holomorphic Legendrian curves can be approximated uniformly by proper, complete curves in convex domains.

Any bordered Riemann surface can be properly embedded as a Legendrian curve in a convex domain.

The results apply to both strongly convex and general convex domains, broadening their scope.

Abstract

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.