Deformations of the gyroid and Lidinoid minimal surfaces

Abstract

The gyroid and Lidinoid are triply periodic minimal surfaces of genus three embedded in R3 that contain no straight lines or planar symmetry curves. They are the unique embedded members of the associate families of the Schwarz P and H surfaces. We prove the existence of two 1-parameter families of embedded triply periodic minimal surfaces of genus three that contain the gyroid and a single 1-parameter family that contains the Lidinoid. We accomplish this by using the flat structures induced by the holomorphic 1-forms G dh, (1/G) dh, and dh. An explicit parametrization of the gyroid using theta functions enables us to find a curve of solutions in a two-dimensional moduli space of flat structures by means of an intermediate value argument.