High genus periodic gyroid surfaces of non-positive Gaussian curvature

TL;DR

This paper introduces a new method for generating complex periodic surfaces with non-positive Gaussian curvature, expanding the known family of gyroid structures with higher genus values and potential minimal surface properties.

Contribution

A novel approach for creating high-genus periodic gyroid surfaces of non-positive Gaussian curvature, including six previously unknown structures and the minimal Schoen-Luzzati gyroid.

Findings

Generated six new high-genus gyroid surfaces

Identified most as minimal surfaces except for genus 21

Reproduced the known Schoen-Luzzati gyroid surface

Abstract

In this paper we present the novel method for the generation of periodic embedded surfaces of nonpositive Gaussian curvature. The structures are related to the local minima of the scalar order parameter Landau-Ginzburg hamiltonan for microemulsions. The method is used to generate six unknown surfaces of Ia$\bar 3$d symmetry (gyroid) of genus 21, 53, 69, 109, 141 and 157 per unit cell. All of them but that of genus 21 are most likely the minimal surfaces. Schoen-Luzzati gyroid minimal surface of genus 5 (per unit cell) is also obtained.