Conformal Structures and Period Matrices of Polyhedral Surfaces

TL;DR

This paper explores the use of linear discrete Riemann surface theory to analyze polyhedral surfaces in three-dimensional space, enabling computation of holomorphic forms and period matrices, including for complex surfaces like Lawson genus-2.

Contribution

It introduces a method to interpret embedded polyhedral surfaces as discrete Riemann surfaces and compute their period matrices, including new results for the Lawson genus-2 surface.

Findings

Successfully computed the period matrix of the Lawson genus-2 surface

Validated numerical methods with known results

Provided a framework for analyzing polyhedral surfaces as discrete Riemann surfaces

Abstract

We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples, recovering known results to test the numerics and giving the yet unknown period matrix of the Lawson genus-2 surface.