The nearly Kähler structure and minimal surfaces in \(S^6\) (Q1389921)

The author studies minimal surfaces in the nearly Kähler 6-sphere \(S^6\). As is well known, the imaginary Cayley numbers can be used to introduce an almost complex structure on the 6-dimensional sphere which is compatible with the standard metric. Besides \(S^2\), \(S^6\) is the only sphere which can admit an almost complex structure. This, of course, makes the study of its submanifolds even more interesting and important. In the paper under review, the author studies minimal surfaces in \(S^6\) which have constant Kähler angle. Special cases of this are complex curves (surfaces for which \(J\) preserves the tangent space) and minimal totally real surfaces (for which \(J\) maps the tangent space into the normal space). It is shown that a complete minimal surface with constant Kähler angle, which is not a complex curve, and with nonnegative Gaussian curvature \(K\), is either totally geodesic or flat, and that in the second case \(M\) is either totally real or superminimal. However, one should note that the definition of superminimal used is not the standard one for minimal surfaces in spheres, but corresponds only to minimal surfaces with ellipse of curvature a circle. How to obtain those with constant Kähler angle is described by \textit{J. Bolton}, \textit{L. Vrancken} and \textit{L. M. Woodward} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 407-427 (1994; Zbl 0861.53061)].