Metrics with four conic singularities and spherical quadrilaterals

TL;DR

This paper classifies spherical quadrilaterals with four conic singularities, focusing on cases where two angles are multiples of pi, linking geometric classification to Heun's equations with specific monodromy properties.

Contribution

It provides a classification of spherical quadrilaterals with four conic singularities up to isometry, especially when two angles are multiples of pi, connecting geometry with Heun's equations.

Findings

Classification up to isometry for specific angle cases

Connection between geometric quadrilaterals and Heun's equations

Insight into monodromy of associated differential equations

Abstract

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four 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 quadrilaterals and perform the classification up to isometry in the case that two angles at the corners are multiples of pi. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy.