Volume 25, issue 2 (2021)

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Quadratic differentials and circle patterns on complex projective tori

Wai Yeung Lam

Geometry & Topology 25 (2021) 961–997

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.

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circle patterns, discrete conformal geometry, complex projective structures
Mathematical Subject Classification 2010
Primary: 52C26
Secondary: 05B40, 30F60, 32G15, 57M50
Received: 5 October 2019
Revised: 10 March 2020
Accepted: 26 April 2020
Published: 27 April 2021
Proposed: David Gabai
Seconded: Mladen Bestvina, Anna Wienhard
Wai Yeung Lam
Mathematics Research Unit
Université du Luxembourg
Mathematics Department
Brown University
Providence, RI
United States
Beijing Institute of Mathematical Sciences and Applications