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 Gdh, (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.

Keywords

minimal surfaces, main/gyroid, lidinoid, triply periodic, flat structures

Mathematical Subject Classification

Primary: 53A10

Secondary: 49Q05, 30F30

Authors