Uniformizable singular projective structures on Riemann surface orbifolds

TL;DR

This paper characterizes complex projective structures on Riemann surface orbifolds that have injective developing maps, exploring their local and global properties and their relevance to uniformization problems in complex geometry.

Contribution

It provides a detailed description of local and global properties of uniformizable projective structures on Riemann surface orbifolds, linking them to bounded covering structures.

Findings

Projective structures are locally bounded.

Global structure characterized for finite type orbifolds.

Connection established with bounded covering projective structures.

Abstract

This paper is devoted to characterizing complex projective structures defined on Riemann surface orbifolds and giving rise to injective developing maps defined on the monodromy covering of the surface (orbifold) in question. The relevance of these structures stems from several problems involving vector fields with uniform solutions as well as from problems about "simultaneous uniformization" for leaves of foliations by Riemann surfaces. In this paper, we first describe the local structure of the mentioned projective structures showing, in particular, that they are locally bounded. In the case of Riemann surface orbifolds of finite type, the previous result will then allow us to provide a detailed global picture of these projective structures by exploiting their connection with the class of "bounded covering projective structures".