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

We prove existence of Schoen's and other triply periodic minimal surfaces via conjugate (polygonal) Plateau problems. The simpler of these minimal surfaces can be deformed into constant mean curvature surfaces by solving analogous Plateau problems in S³. The required contours in S³ are obtained by working with the great circle orbits of Hopf S¹-actions in the same way as with families of parallel lines in ³. Annular Plateau problems give new embedded minimal surfaces in S³. For many of the minimal surfaces in ³ global Weierstraß representations are derived.Dedicated to Wilhelm Klingenberg