Existence of the tetragonal and rhombohedral deformation families of the gyroid

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.