Discrete Riemann surfaces: Linear discretization and its convergence

Alexander Bobenko; Mikhail Skopenkov

doi:10.1515/crelle-2014-0065

Discrete Riemann surfaces: Linear discretization and its convergence

Alexander Bobenko, Mikhail Skopenkov

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	217-250
Number of pages	34
Journal	Journal fur die Reine und Angewandte Mathematik
Volume	720
Issue number	720
DOIs	https://doi.org/10.1515/crelle-2014-0065
State	Published - Aug 19 2014
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-03
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1515/crelle-2014-0065

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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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ER -

Discrete Riemann surfaces: Linear discretization and its convergence

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Bibliographical note

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