Prescribing the Preschwarzian in several complex variables

TL;DR

This paper generalizes the Schwarzian operator to several complex variables, solving related equations and defining transformations to analyze univalence and invariance properties in higher dimensions.

Contribution

It introduces a new operator equation in several complex variables and explores the properties and transformations of functions under this operator, extending classical one-variable results.

Findings

Solved the several complex variables preSchwarzian operator equation.

Defined a transformation $f_eta$ based on the operator equation.

Analyzed properties like M"{o}bius invariance and univalence in higher dimensions.

Abstract

We solve the several complex variables preSchwarzian operator equation $[Df(z)]^{-1}D^2f(z)=A(z)$, $z\in \C^n$, where $A(z)$ is a bilinear operator and $f$ is a $\C^n$ valued locally biholomorphic function on a domain in $\C^n$. Then one can define a several variables $f\to f_\alpha$ transform via the operator equation $[Df_\alpha(z)]^{-1}D^2f_\alpha(z)=\alpha[Df(z)]^{-1}D^2f(z)$, and thereby, study properties of $f_\alpha$. This is a natural generalization of the one variable operator $f_\alpha(z)$ in \cite{DSS66} and the study of its univalence properties, e.g., the work of Royster \cite{Ro65} and many others. M\"{o}bius invariance and the multivariables Schwarzian derivative operator of T. Oda \cite{O} play a central role in this work.