Proper holomorphic Legendrian curves in $SL_2(\mathbb{C})$

TL;DR

This paper proves that every open Riemann surface can be properly embedded as a holomorphic Legendrian curve in $SL_2( olinebreak bC)$, leading to new examples of flat fronts in hyperbolic space with arbitrary complex structures.

Contribution

It establishes the existence of proper holomorphic Legendrian embeddings of all open Riemann surfaces into $SL_2( olinebreak bC)$, a novel result in complex contact geometry.

Findings

Every open Riemann surface properly embeds in $SL_2(bC)$ as a Legendrian curve.

Existence of proper, weakly complete, flat fronts in $bH^3$ with arbitrary complex structure.

Extension of complex geometric methods to hyperbolic geometry applications.

Abstract

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts in the real hyperbolic space $\mathbb{H}^3$ with arbitrary complex structure.