On metrics of curvature $1$ with four conic singularities on tori and on the sphere

Abstract

We discuss conformal metrics of curvature $1$ on tori and on the sphere, with four conic singularities whose angles are multiples of $\pi$. Besides some general results we study in detail the family of such symmetric metrics on the sphere, with angles $(\pi,3\pi,\pi,3\pi)$. As a consequence we find new Heun’s equations whose general solution is algebraic.