Transforms for minimal surfaces in the 5-sphere (Q1001996)

The authors provide a nice procedure to define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere. This procedure permits to construct a sequence of minimal immersions into the 5-sphere. The angle \(\theta\) determined by the ratio of the lengths of the minor and major axes of de ellipse of curvature \(E,\) and the unit vector \(N\) in the normal space orthogonal to the plane containing \(E\) play an important role in this construction. If \(R_{\theta}\) is the rotation of the normal space through angle \(\theta\) about the minor axis of \(E\), then the transforms are obtained applying \(R_{\pm\theta}\) to \(\pm N.\) The various choices available of sign and orientation give two essentially different transforms. These transformed surfaces are also minimal, and the two transforms are mutual inverses. This permits to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of the complex projective 3-space, which gives the converse of a construction considered in a previous paper [\textit{J. Bolton} and \textit{L. Vrancken}, Asian J. Math. 9, No. 1, 45--55 (2005; Zbl 1079.53086)], and illustrate this explicitly in the case of bipolar minimal surfaces in the 5-sphere.