Monodromies of Projective Structures on Surface of Finite-type

TL;DR

This paper characterizes the monodromies of projective structures with fuchsian-type singularities on finite-type surfaces, linking representations of the fundamental group to branched projective structures and exploring their deformations.

Contribution

It provides a geometric and topological classification of local conical projective structures with simple pole Schwarzian derivatives and studies their isomonodromic deformations.

Findings

Any representation of the fundamental group can be realized as a holonomy of a branched projective structure.

Classification of local conical structures with simple pole Schwarzian derivatives.

Analysis of isomonodromic deformations and angle minimization problems.

Abstract

We characterize the monodromies of projective structures with fuchsian-type singularities. Namely, any representation from the fundamental group of a Riemann surface of finite-type in $PSL_2(\mathbb{C})$ can be represented as the holonomy of branched projective structure with fuchsian-type singularities over the cusps. We made a geometrical/topological study of all local conical projective structures whose Schwarzian derivative admits a simple pole at the cusp. Finally, we explore isomonodromic deformations of such projective structures and the problem of minimizing angles.