Spherical rectangles

TL;DR

This paper classifies spherical quadrilaterals with specific angle conditions and relates them to Heun equations with real parameters, providing a finite enumeration of families and insights into their conformal moduli and monodromy.

Contribution

It offers a classification of spherical quadrilaterals with angles as odd multiples of pi/2 and connects this to the accessory parameter problem for Heun's equations with real parameters.

Findings

Finite number of continuous families for given angles.

Conformal modulus is either bounded above or below within each family.

Classification applies to Heun's equations with unitary monodromy.

Abstract

We study spherical quadrilaterals whose angles are odd multiples of pi/2, and the equivalent accessory parameter problem for the Heun equation. We obtain a classification of these quadrilaterals up to isometry. For given angles, there are finitely many one-dimensional continuous families which we enumerate. In each family the conformal modulus is either bounded from above or bounded from below, but not both, and the numbers of families of these two types are equal. The results can be translated to classification of Heun's equations with real parameters, whose exponent differences are odd multiples of 1/2, with unitary monodromy.