The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions (Q1263087)

The existence of certain triply periodic minimal surfaces in \({\mathbb{R}}^ 3\) previously obtained by A. Schoen (and of more such surfaces) is proved by considering conjugate polygonal Plateau problems. Deformations of these to triply periodic constant mean curvature surfaces are obtained by solving a conjugate Plateau problem in \(S^ 3\). The corresponding polygons in \(S^ 3\) have edges on parallel geodesics where the Euclidean polygons have parallel edges. Also new compact embedded minimal surfaces in \(S^ 3\) are obtained by this method. For many of the Euclidean minimal surfaces explicit Weierstrass representations are derived.