On the harmonic M\"obius transformations

TL;DR

This paper characterizes the relationship between two locally univalent harmonic mappings that share the same harmonic Schwarzian derivative, extending classical results from analytic functions to harmonic mappings.

Contribution

It provides a complete characterization of when two harmonic mappings with equal harmonic Schwarzian derivatives are related by a M"obius transformation.

Findings

Harmonic Schwarzian derivatives uniquely determine harmonic mappings up to M"obius transformations.

Extension of classical analytic results to harmonic mappings.

Complete description of the relationship between harmonic mappings with equal harmonic Schwarzian derivatives.

Abstract

It is well-known that two locally univalent analytic functions $\varphi$ and $\psi$ have equal Schwarzian derivative if and only if there exists a non-constant M\"obius transformation $T$ such that $\varphi=T\circ \psi$. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schwarzian derivative.