The convergence of discrete period matrices

TL;DR

This paper proves that discrete models of polyhedral surfaces with quadrilateral decompositions converge to their continuous counterparts in terms of energy, period matrices, and Abelian integrals, under certain geometric conditions.

Contribution

It establishes convergence of discrete harmonic differentials, period matrices, and Abelian integrals on polyhedral surfaces with quadrilateral decompositions, including improved rates with adapted quadrangulations.

Findings

Discrete Dirichlet energies converge to continuous energies.

Discrete period matrices converge to continuous period matrices.

Convergence of Abelian integrals when diagonals are orthogonal.

Abstract

We study compact polyhedral surfaces as Riemann surfaces and their discrete counterparts obtained through quadrilateral cellular decompositions and a linear discretization of the Cauchy-Riemann equation. By ensuring uniformly bounded interior and intersection angles of diagonals, we establish the convergence of discrete Dirichlet energies of discrete harmonic differentials with equal black and white periods to the Dirichlet energy of the corresponding continuous harmonic differential with the same periods. This convergence also extends to the discrete period matrix, with a description of the blocks of the complete discrete period matrix in the limit. Moreover, when the quadrilaterals have orthogonal diagonals, we observe convergence of discrete Abelian integrals of the first kind. Adapting the quadrangulations around conical singularities allows us to improve the convergence rate to a linear function of the maximum edge length.