Quadratic differentials and circle patterns on complex projective tori

Abstract

Given a triangulation of a closed surface, we consider a cross-ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross-ratio system induces a complex projective structure together with a circle pattern. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross-ratio systems with prescribed Delaunay angles to the Teichmüller space of the closed torus is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.