Flat cone metric space in terms of Schwarz-Christoffel maps

In this talk, we propose a complex analytic approach to the description of moduli space of flat metrics on a sphere with cone singularities. Namely, we consider the Schwarz-Christoffel map that sends a punctured upper half plane to a curvilinear polygon that shall be glued to form a surface with cone singularities. We discuss the relation with the monodromy invariant Hermitian form for Lauricella hypergeometric functions and the cone metric space as a hyperbolic space.