Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere

TL;DR

This paper proves that discrete harmonic and holomorphic functions on triangulated ramified coverings of the Riemann sphere converge to their continuous analogs, with error estimates depending linearly on the maximal edge length.

Contribution

It introduces a method to discretize and analyze harmonic and holomorphic functions on ramified coverings, establishing convergence and error bounds.

Findings

Discrete period matrices converge to continuous ones.

Error estimates are linear in the maximal edge length.

Convergence of discrete holomorphic integrals is established.

Abstract

We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat{\mathbb{C}}$. Based on a triangulation of this covering of the sphere $\mathbb{S}^2\cong \hat{\mathbb{C}}$ and its stereographic projection, we define discrete (multi-valued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.