Metrics with conic singularities and spherical polygons

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.