Discrete Riemann surfaces: linear discretization and its convergence

TL;DR

This paper introduces a linear discretization method for complex analysis on Riemann surfaces, proving its convergence and establishing a discrete Riemann--Roch theorem using energy estimates.

Contribution

It develops a new linear discretization approach for complex analysis on Riemann surfaces and proves key convergence results and a discrete Riemann--Roch theorem.

Findings

Discrete period matrices converge to continuous ones

Discrete Abelian integrals converge to continuous integrals

Established a discrete Riemann--Roch theorem

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.