Schwarzian derivatives for pluriharmonic mappings

TL;DR

This paper introduces and analyzes Schwarzian derivatives for pluriharmonic mappings in several complex variables, establishing fundamental properties, characterizations, and differences from the planar case.

Contribution

It defines pre-Schwarzian and Schwarzian derivatives for pluriharmonic mappings in ${f C}^n$, proving key properties and exploring their implications and limitations.

Findings

Pre-Schwarzian is stable only under rotations.

Vanishing Schwarzian implies the analytic part is a M"obius transformation.

Differences between planar and higher-dimensional theories are highlighted.

Abstract

A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a M\"obius transformation. Some observations are made related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations, revealing differences between the theories in the plane and in higher dimensions. An example is given that rules out the possibility for a shear construction theorem to hold in ${\mathbb C}^n$, for $n\geq2$.