The gyroid is embedded and has constant mean curvature companions (Q1815286)

The gyroid is a triply period minimal surface originally found by \textit{A. Schoen} [`Infinite periodic minimal surfaces without self-intersections', NASA Technical Note TN D-5541 (1970); see also `Geometric analysis and computer graphics 1988', 147-158 (1991; Zbl 0776.68009)]. It belongs to the associated family of Schwarz' \(P\)- and \(D\)-surface. The gyroid was a candidate of the first embedded complete minimal surface which contains no reflectional symmetry lines. The authors prove embeddedness of the gyroid, as well as embeddedness of a similar surface \(HG\) in the associate family of Schwarz' \(H\)-surface found by Lidin and Larsson. Using results of Ross and Kapouleas, the authors obtain triply periodic embedded constant mean curvature gyroids.