Quadratic differentials and circle patterns on complex projective tori

Lam, Wai Yeung¹

¹ Mathematics Research Unit, Université du Luxembourg, Esch-sur-Alzette, Luxembourg, Mathematics Department, Brown University, Providence, RI, United States, Beijing Institute of Mathematical Sciences and Applications, Beijing, China

Geometry & topology, Tome 25 (2021) no. 2, pp. 961-997.

Résumé

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.

Détail

DOI : 10.2140/gt.2021.25.961

Classification : 52C26, 05B40, 30F60, 32G15, 57M50
Keywords: circle patterns, discrete conformal geometry, complex projective structures

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