Deformations of the gyroid and lidinoid minimal surfaces

Adam G. Weyhaupt

Vol. 235 (2008), No. 1, 137–171

DOI: 10.2140/pjm.2008.235.137

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 2000

Primary: 53A10

Secondary: 49Q05, 30F30

Milestones

Received: 31 May 2007

Revised: 6 July 2007

Accepted: 9 July 2007

Published: 1 March 2008