Metrics with conic singularities and spherical polygons

TL;DR

This paper classifies spherical polygons with conic singularities, linking geometric properties to differential equations with regular singularities and unitary monodromy, advancing understanding of spherical metrics with prescribed boundary angles.

Contribution

It provides a classification and enumeration of spherical polygons with specific angle conditions, connecting geometric structures to differential equations with regular singularities.

Findings

Classification of spherical polygons with two non-multiple pi angles

Enumeration results for these polygons

Connection between geometric structures and differential equations

Abstract

A spherical n-gon is a bordered surface homeomorphic to a closed disk, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of pi. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.