Schwarzians on the Heisenberg group

TL;DR

This paper explores different notions of the Schwarzian derivative for contact mappings in the Heisenberg group, introducing new definitions, analyzing their properties, and studying related PDEs and harmonic mappings in this non-Euclidean setting.

Contribution

It introduces two new definitions of the Schwarzian derivative in the Heisenberg group and investigates their properties, kernels, cocycle conditions, and related harmonic mappings.

Findings

Characterization of contact conformal vector fields.

Introduction of Preschwarzian for $ ext{H}_1$.

Analysis of subelliptic PDEs related to the horizontal Jacobian.

Abstract

We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group $\mathbb{H}_1$ and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) the classical type Schwarzian refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Our auxiliary results include a characterization of the contact conformal vector fields. Inspired by ideas of Chuaqui--Duren--Osgood, Hern\'andez, Mart\'in and Venegas, we introduce the Preschwarzian for mappings in $\mathbb{H}_1$. Furthermore, we study results in the theory of subelliptic PDEs for the horizontal Jacobian and related differential expressions for harmonic mappings and the gradient harmonic mappings, the latter notion introduced here in the setting of $\mathbb{H}_1$.