TOTALLY REAL MINIMAL SURFACES WITH NON-CIRCULAR ELLIPSE OF CURVATURE IN THE NEARLY KÄHLER S⁶

Abstract

In [2] we discussed almost complex curves in the nearly Kähler S⁶. These are surfaces with constant Kähler angle 0 or π and, as a consequence of this, are also minimal and have circular ellipse of curvature. We also considered minimal immersions with constant Kähler angle not equal to 0 or π, but with ellipse of curvature a circle. We showed that these are linearly full in a totally geodesic S⁵ in S⁶ and that (in the simply connected case) each belongs to the S¹-family of horizontal lifts of a totally real (non-totally geodesic) minimal surface in [Copf ]P². Indeed, every element of such an S¹-family has constant Kähler angle and in each family all constant Kähler angles occur. In particular, every minimal immersion with constant Kähler angle and ellipse of curvature a circle is obtained by rotating an almost complex curve which is linearly full in a totally geodesic S⁵.