# Existence of the tetragonal and rhombohedral deformation families of the gyroid

## TL;DR

This paper proves the existence of two new families of triply periodic minimal surfaces with specific symmetries, extending previous local existence results to more general torus configurations.

## Contribution

It extends the existence proof of gyroid-related minimal surfaces from rectangular to more general branched tori, broadening the understanding of their geometric families.

## Key findings

- Established the existence of tG and rGL families with specific symmetries.
- Extended the proof technique to non-rectangular branched tori.
- Confirmed the inclusion of gyroid within both families.

## Abstract

We provide an existence proof for two 1-parameter families of embedded triply periodic minimal surfaces of genus three, namely the tG family with tetragonal symmetry that contains the gyroid, and the rGL family with rhombohedral symmetry that contains the gyroid and the Lidinoid, both discovered numerically in the 1990s. The existence was previously proved within a neighborhood of the gyroid and the Lidinoid, using Weierstrass data defined on branched rectangular tori. Our main contribution is to extend the technique to branched tori that are not necessarily rectangular.