Hyperbolic complex contact structures on $\mathbb{C}^{2n+1}$

TL;DR

This paper constructs specific complex contact structures on complex Euclidean spaces that restrict the existence of non-trivial holomorphic Legendrian curves, demonstrating these structures differ fundamentally from the standard contact structure.

Contribution

It introduces new complex contact structures on b^{2n+1} with unique properties, expanding understanding of contact geometry in complex Euclidean spaces.

Findings

Holomorphic Legendrian maps b o b^{2n+1} are necessarily constant.

Constructed contact structures are not globally contactomorphic to the standard structure.

Demonstrated existence of complex contact structures with restricted Legendrian curves.

Abstract

In this paper we construct complex contact structures on $\mathbb{C}^{2n+1}$ for any $n\ge 1$ with the property that every holomorphic Legendrian map $\mathbb{C}\to \mathbb{C}^{2n+1}$ is constant. In particular, these contact structures are not globally contactomorphic to the standard complex contact structure on $\mathbb{C}^{2n+1}$.