Dihedral monodromy of cone spherical metrics

TL;DR

This paper characterizes the set of conical angles for cone spherical metrics with dihedral monodromy on punctured Riemann surfaces, using recent advances in quadratic differentials.

Contribution

It provides a complete characterization of conical angles for metrics with dihedral monodromy, connecting local invariants of quadratic differentials.

Findings

Complete set of conical angles characterized

Connection established between monodromy and quadratic differentials

Advances in understanding cone spherical metrics with dihedral symmetry

Abstract

Among metrics of constant positive curvature on a punctured compact Riemann surface with conical singularities at the punctures, dihedral monodromy means that the action of the monodromy group globally preserves a pair of antipodal points. Using recent results about local invariants of quadratic differentials, we give a complete characterization of the set of conical angles realized by some cone spherical metric with dihedral monodromy.