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Discrete Riemann surfaces: Linear discretization and its convergence

Alexander Bobenko and Mikhail Skopenkov

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.

Funding source: Deutsche Forschungsgemeinschaft

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

Funding statement: 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.

Acknowledgements

The authors are grateful to D. Chelkak, S. von Deylen, I. Dynnikov, A. Gaifullin, F. Günther, S. Lando, C. Mercat, A. Pakharev, M. Wardetzky for useful discussions and to S. Tikhomirov for writing a software for numerical experiments.

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Received: 2013-11-26
Revised: 2014-4-25
Published Online: 2014-8-19
Published in Print: 2016-11-1

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