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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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1435-5345
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Volume 2016, Issue 720

# Discrete Riemann surfaces: Linear discretization and its convergence

Alexander Bobenko
/ Mikhail Skopenkov
• Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny per. 19, 127994 Moscow; and Higher School of Economics, Vavilova Str. 7, 117312 Moscow, Russian Federation
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Published Online: 2014-08-19 | DOI: https://doi.org/10.1515/crelle-2014-0065

## Abstract

We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann–Roch theorem. The proofs use energy estimates inspired by electrical networks.

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## About the article

Revised: 2014-04-25

Published Online: 2014-08-19

Published in Print: 2016-11-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Collaborative Research Center SFB/TR 109 Discretization in Geometry and Dynamics

The first author was partially supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”. The second author was partially supported by the President of the Russian Federation grant MK-5490.2014.1, by “Dynasty” foundation, and by the Simons–IUM fellowship. Part of the work on this paper was done during the stay of the second author at King Abdullah University of Science and Technology in Saudi Arabia.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 720, Pages 217–250, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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