Metrics with conic singularities and spherical polygons

Abstract

A spherical nn-gon is a bordered surface homeomorphic to a closed disk, with nn distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 11, 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 ππ. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.