Quadratic differentials and circle patterns on complex projective tori

TL;DR

This paper explores the relationship between cross ratio systems, circle patterns, and complex projective structures on triangulated tori, revealing a covering map property with implications for discrete holomorphic quadratic differentials.

Contribution

It demonstrates that the projection from cross ratio systems with fixed Delaunay angles to Teichmüller space is a covering map with at most one branch point for triangulated tori.

Findings

Projection is a covering map with at most one branch point.

Introduces a notion of discrete holomorphic quadratic differentials.

Establishes a link between circle patterns and complex projective structures.

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 on the closed surface. 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 Teichm\"{u}ller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.