Uniformizable singular projective structures on Riemann surface orbifolds

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".