Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere
Provides a theoretical foundation for numerical computation of period matrices and holomorphic integrals on a class of Riemann surfaces, but the result is incremental as it extends known convergence results to ramified coverings with specific triangulation adaptations.
The paper proves that discrete period matrices and discrete holomorphic integrals, defined via adapted triangulations of ramified coverings of the Riemann sphere, converge to their continuous counterparts with an error linear in the maximal edge length.
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.