# Darboux charts around holomorphic Legendrian curves and applications

## TL;DR

This paper develops holomorphic Darboux charts around Legendrian curves and submanifolds in complex contact manifolds, enabling approximation and embedding results for Legendrian immersions and embeddings.

## Contribution

It introduces a holomorphic Darboux chart around Legendrian curves and submanifolds, facilitating approximation and embedding theorems in complex contact geometry.

## Key findings

- Holomorphic Darboux charts exist around Legendrian curves and isotropic Stein submanifolds.
- Any Legendrian immersion from an open Riemann surface can be approximated by embeddings.
- Compact Legendrian immersions can be approximated by complete embeddings.

## Abstract

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold.