Holomorphic Legendrian curves

TL;DR

This paper develops approximation and desingularization techniques for holomorphic Legendrian curves in complex contact structures, proving existence and embedding theorems for such curves on various Riemann surfaces and complex manifolds.

Contribution

It introduces new methods to construct proper and complete holomorphic Legendrian embeddings, solving open problems in the theory of Legendrian curves in complex contact manifolds.

Findings

Every open Riemann surface admits a proper holomorphic Legendrian embedding into complex Euclidean space.

Existence of topological and complete Legendrian embeddings for bordered Riemann surfaces.

Every complex contact manifold contains relatively compact, complete Legendrian curves.

Abstract

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M\hookrightarrow\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface $M=\mathring M\cup bM$ there exists a topological embedding $M\hookrightarrow \mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding $\mathring M\hookrightarrow \mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$.