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The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two. (English) Zbl 0827.53047

The authors determine the structure of complete minimal real Kaehler submanifolds in codimension 2. These submanifolds are essentially holomorphic vector bundles over Riemann surfaces, and the immersion can be explicitly parametrized in terms of a Weierstrass-type representation which leads to the construction of complete examples.
Reviewer: E.Hoy (Friedberg)

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

[1] [Br] Bryant, R.: Some remarks on the geometry of austere manifolds. Bol. Soc. Bras. Mat.21, 133-157 (1991) · Zbl 0760.53034 · doi:10.1007/BF01237361
[2] [Ch] Chen, C. C.: The generalized curvature ellipses and minimal surfaces. Bull. Acad. Sinica11, 329-336 (1983) · Zbl 0526.53051
[3] [D-G1] Dajczer, M., Gromoll, D.: Real Kaehler submanifolds and uniqueness of the Gauss map. J. Diff. Geom.22, 13-28 (1985) · Zbl 0587.53051
[4] [D-G2] Dajczer, M., Gromoll, D.: Rigidity of complete Euclidean hypersurfaces. J. Diff. Geom.31, 401-416 (1990) · Zbl 0667.53003
[5] [D-R1] Dajczer, M., Rodríguez, L.: Rigidity of real Kaehler submanifolds. Duke Math. J.53, 211-220 (1986) · Zbl 0599.53005 · doi:10.1215/S0012-7094-86-05314-7
[6] [D-R2] Dajczer, M., Rodríguez, L.: Complete real Kähler minimal submanifolds. J. reine angew. Math.419, 1-8 (1991) · Zbl 0726.53041
[7] [Fu] Furuhata, H.: Construction and classification of isometric minimal immersions of Kähler manifolds into Euclidean spaces. Preprint.
[8] [H-L] Harvey, R., Lawson, B.: Calibrated geometries. Acta Math148, 47-157 (1982) · Zbl 0584.53021 · doi:10.1007/BF02392726
[9] [H-O] Hoffman, D., Osserman, R.: The geometry of the generalized Gauss map. Memoris Am. Math. Soc.28 (1980)
[10] [La] Lawson, B., Lectures on minimal surfaces. Publish or Perish Inc. 1980
[11] [R-T] Rigoli, M., Tribuzy, R.: The Gauss map for Kählerian submanifolds ofR n . Trans. Am. Math. Soc.332, 515-530 (1992) · Zbl 0760.53004 · doi:10.2307/2154180
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