The existence of real nine-dimensional manifolds which include classical one-parameter families of triply periodic minimal surfaces

Abstract

Triply periodic minimal surfaces have been studied in many fields of natural science, and in particular, many one-parameter families of triply periodic minimal surfaces of genus three have been considered. In 1990s, the moduli theory of triply periodic minimal surfaces established by C. Arezzo and G. P. Pirola [1], [14], and they studied a relationship between the nullity of a minimal surface and the differential of its real period map from the viewpoint of complex geometry. The present paper develops their theory in terms of a real differential geometric aspect, and, by applying the classical transversal property to the real period map, we obtain the numerical evidence for the existence of real nine-dimensional manifolds of triply periodic minimal surfaces which include such one-parameter families. For each case that the transversal property fails, we give values of parameters from which new one-parameter families of triply periodic minimal surfaces issue.