# Complex projective structures with maximal number of M\"obius transformations

## TL;DR

This paper characterizes complex projective structures on Riemann surfaces with the maximum number of automorphisms, identifying them as Fuchsian uniformizations of Hurwitz surfaces and linking Galois Bely
2B curves to unique invariant structures.

## Contribution

It establishes a precise link between maximal automorphism groups of projective structures and Fuchsian uniformizations of Hurwitz surfaces, and characterizes Galois Bely
2B curves via invariance properties.

## Key findings

- Maximal automorphism groups correspond to Fuchsian uniformizations of Hurwitz surfaces.
- Galois Bely
2B curves have a unique invariant projective structure under all biholomorphisms.
- Characterization of structures with maximal symmetry in terms of classical uniformizations.

## Abstract

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely\u{\i} curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.