On classical uniformization theorems for higher dimensional complex Kleinian groups

TL;DR

This paper demonstrates that classical uniformization theorems like Bers' and Koebe's do not extend to higher-dimensional complex Kleinian groups acting on complex projective space.

Contribution

It provides counterexamples showing the failure of these classical theorems in higher dimensions, highlighting limitations of existing uniformization results.

Findings

Bers' simultaneous uniformization fails in higher dimensions

Koebe's retrosection theorem does not hold for higher-dimensional groups

Classical uniformization theorems are not universally applicable in complex projective spaces

Abstract

In this article we show that Bers' simultaneous uniformization as well as the K\"oebe's retrosection theorem are not longer true for discrete groups of projective transformations acting on the complex projective space.