, Volume 64, Issue 3, pp 291-357

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

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Abstract

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 ℝ3. Annular Plateau problems give new embedded minimal surfaces in S3. For many of the minimal surfaces in ℝ3 global Weierstraß representations are derived.

Dedicated to Wilhelm Klingenberg