Discrete minimal surfaces: Old and New

TL;DR

This paper surveys structure-preserving discretizations of minimal surfaces, focusing on a circle pattern-based approach that connects to Teichmüller theory and explores various modifications and open questions.

Contribution

It introduces a discretization method via circle patterns, linking discrete minimal surfaces to classical Teichmüller theory and exploring various modifications.

Findings

All simply connected discrete minimal surfaces of this type can be constructed from circle patterns.

The discrete Weierstrass representation links minimal surfaces to circle pattern deformations.

Variants include modifications of mean curvature and alternative discrete conformal structures.

Abstract

We survey structure-preserving discretizations of minimal surfaces in Euclidean space. Our focus is on a discretization defined via parallel face offsets of polyhedral surfaces, which naturally leads to a notion of vanishing mean curvature and a corresponding variational characterization. All simply connected discrete minimal surfaces of this type can be constructed from circle patterns via a discrete Weierstrass representation formula. This representation links the space of discrete minimal surfaces to the deformation space of circle patterns, and thereby to classical Teichm\"uller theory. We also discuss variants of discrete minimal surfaces obtained by modifying the definition of mean curvature, restricting the variational criterion, or replacing circle pattern data with discrete conformal equivalence, Koebe-type circle packings, or quadrilateral meshes with factorized cross ratios. We conclude with open questions on discrete minimal surfaces.