# Co-axial monodromy

## TL;DR

This paper characterizes the possible angles at conic singularities for constant positive curvature metrics on punctured spheres with co-axial monodromy, completing prior classifications and solving related potential multiplicity problems.

## Contribution

It provides a complete description of singularity angles and multiplicities for specific Riemannian metrics with co-axial monodromy, extending recent results.

## Key findings

- Complete classification of angles at conic singularities.
- Solution to the multiplicity problem of critical points of logarithmic potentials.
- Extension of previous work by Mondello and Panov.

## Abstract

For Riemannian metrics of constant positive curvature on a punctured sphere with conic singularities at the punctures and co-axial monodromy of the developing map, possible angles at the singularities are completely described. This completes the recent result of Mondello and Panov. The related problem of describing possible multiplicities of critical points of logarithmic potentials of finitely many charges is also solved.