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Journal of the London Mathematical Society 1997 56(3):625-644; doi:10.1112/S0024610797005541
© 1997 by London Mathematical Society
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© The London Mathematical Society

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kähler S6

John Bolton, Luc Vrancken and Lyndon M. Woodward

Department of Mathematical Sciences, University of Durham South Road, Durham DH1 3LE
Departement Wiskunde, Katholieke Universiteit Leuven Celestijnenlaan 200 B, B-3001 Leuven, Belgium

Received 4 September 1995.

In [2] we discussed almost complex curves in the nearly Kähler S6. These are surfaces with constant Kähler angle 0 or {pi} 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 {pi}, 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 CP2. 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.


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