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Quadratic differentials
and circle patterns on complex projective tori
Wai Yeung Lam |
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Geometry & Topology 25 (2021) 961–997
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Abstract |
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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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Keywords
circle patterns, discrete conformal geometry, complex
projective structures
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Mathematical Subject Classification 2010
Primary: 52C26
Secondary: 05B40, 30F60, 32G15, 57M50
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References |
Publication
Received: 5 October 2019
Revised: 10 March 2020
Accepted: 26 April 2020
Published: 27 April 2021
Proposed: David Gabai
Seconded: Mladen Bestvina, Anna Wienhard
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Authors |
| Wai Yeung Lam | ||
| 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 |
