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From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space

Published online by Cambridge University Press:  17 April 2009

J. Bolton ,
C. Scharlach  and
L. Vrancken
J. Bolton
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom e-mail: john.bolton@dur.ac.uk
C. Scharlach
Affiliation:
Fachbereich Mathematik, Sekr. MA 8–3, Technische Universität Berlin, Strasse des 17 Juni 136, 10623 Berlin, Germany e-mail: cs@sfb288.math.tu-berlin.de
L. Vrancken
Affiliation:
Laboratoire de Mathématiques, LAMATH, ISTV2, Université de Valenciennes59313 Valenciennes Cedex 9, France e-mail: luc.vrancken@univ-valenciennes.fr
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In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 3 , December 2002 , pp. 465 - 475
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Blair, D.E., Dillen, F., Verstraelen, L. and Vrancken, L., ‘Calabi curves as holomorphic Legendre curves and Chen's inequality’, Kyungpook Math. J. 35 (1996), 407–416.Google Scholar
[2]Bolton, J., Scharlach, C., Vrancken, L. and Woodward, L.M., ‘From certain minimal Lagrangian submanifolds of the 3-dimensional complex projective space to minimal surfaces in the 5-sphere’ (to appear),in Proceedings of the Fifth Pacific Rim Geometry Conference,Tohoku University to appear.Google Scholar
[3]Bolton, J. and Woodward, L.M., ‘Congruence theorems for harmonic maps from a Riemann surface into ℂP n and Sn’, J. London Math. Soc. (2) 45 (1992), 363–376.CrossRefGoogle Scholar
[4]Bryant, R.L.., ‘Conformal and minimal immersions of compact surfaces into the 4-sphere’, J. Differential Geom. 17 (1982), 455–473.CrossRefGoogle Scholar
[5]Bryant, R.L.., ‘Second order families of special Lagrangian 3-folds’, (preprint).Google Scholar
[6]Chen, B.-Y.., ‘Some pinching and classification theorems for minimal submanifolds’, Arch. Math. 60 (1993), 568–578.CrossRefGoogle Scholar
[7]Chen, B.-Y., Dillen, F., Verstraelen, L. and Vrancken, L., ‘An exotic totally real minimal immersion of S 3 in ℂP 3 and its characterization’, Proc. Royal Soc. Edinburgh 126 (1996), 153–165.CrossRefGoogle Scholar
[8]Chen, B.-Y., Dillen, F., Verstraelen, L. and Vrancken, L., ‘Totally real submanifolds of ℂPn satisfying a basic equality’, Arch. Math. 63 (1994), 553–564.CrossRefGoogle Scholar
[9]Reckziegel, H., ‘Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion’, in Global Differential Geometry and Global Analysis (1984), Lecture Notes in Mathematics 1156 (Springer Verlag, Berlin, Heidelberg, New York, 1985), pp. 264–279.CrossRefGoogle Scholar