Abstract
The purpose of this chapter is to present several results on isometric immersions of Kaehler manifolds into real space forms. In fact, most of the results are about real Kaehler submanifolds. By a real Kaehler submanifold \(f\colon M^{2n}\to \mathbb {R}^m\) we mean an isometric immersion of a Kaehler manifold M 2n of complex dimension n ≥ 2 into Euclidean space.
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References
Arezzo, C., Pirola, G., Solci, M.: The Weierstrass representation for pluriminimal submanifolds. Hokkaido Math. J. 33, 357–367 (2004)
Borisenko, A.: Immersion of Kähler manifolds in the class of convex submanifolds. Math. Notes 100, 526–530 (2016)
Burstall, F., Eschenburg, J., Ferreira, M., Tribuzy, R.: Kähler submanifolds with parallel pluri-mean curvature. Differ. Geom. Appl. 20, 47–66 (2004)
Calabi, E.: Isometric embeddings of complex manifolds. Ann. Math. 58, 1–23 (1953)
Calabi, E.: Quelques applications de l’analyse complexe aux surfaces d’aire minima. Topics in Complex Manifolds. University of Montreal, Montreal, pp. 59–81 (1968)
Carvalho, A., Chion, S. and Dajczer, M., Holomorphicity of real Kaehler submanifolds. Preprint (2019)
Dajczer, M., Florit, L.: The holomorphic Gauss parametrization. Manuscripta Math. 129, 127–135 (2009)
Dajczer, M., Gromoll, D.: Real Kaehler submanifolds and uniqueness of the Gauss map. J. Differ. Geom. 22, 13–28 (1985)
Dajczer, M., Gromoll, D.: The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two. Invent. Math. 119, 235–242 (1995)
Dajczer, M., Gromoll, D.: Complete minimal Kaehler surfaces in \(\mathbb {R}^6\). Ann. Global Anal. Geom. 15, 539–541 (1997)
Dajczer, M., Gromoll, D.: Real Kaehler submanifolds in low codimension. Differ. Geom. Appl. 7, 389–395 (1997)
Dajczer, M., Rodríguez, L.: Rigidity of real Kaehler submanifolds. Duke Math. J. 53, 211–220 (1986)
Dajczer, M., Rodríguez, L.: Complete real Kähler submanifolds. J. Reine Angew. Math. 419, 1–8 (1991)
Dajczer, M., Rodríguez, L.: Euclidean hypersurfaces which make a constant angle. In: Differential Geometry. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52, pp. 103–110. Longman Scientific & Technical, Harlow (1991)
Dajczer, M., Rodríguez, L.: On isometric immersions into complex space forms. Math. Ann. 299, 223–230 (1994)
Dajczer, M., Vlachos, Th.: Kaehler submanifolds of hyperbolic space (2018). Preprint
Di Scala, A.: Minimal immersions of Kähler manifolds into Euclidean spaces. Bull. Lond. Math. Soc. 35, 825–827 (2003)
Eschenburg, J.-H., Ferreira, M., Tribuzy, R.: Isotropic ppmc immersions. Differ. Geom. Appl. 25, 351–355 (2007)
Ferreira, M., Tribuzy, R.: Codimension two Kähler submanifolds of space forms. Arch. Math. 79, 520–528 (2002)
Ferus, D.: Immersions with parallel second fundamental form. Math. Z. 140, 87–93 (1974)
Ferus, D.: Symmetric submanifolds of Euclidean space. Math. Ann. 247, 81–93 (1980)
Florit, L., Zheng, F.: A local and global splitting result for real Kaehler Euclidean submanifolds. Arch. Math. 84, 88–95 (2005)
Florit, L., Zheng, F.: Complete real Kaehler Euclidean hypersurfaces are cylinders. Ann. Inst. Four. 57, 155–161 (2007)
Florit, L., Zheng, F.: Complete real Kaehler submanifolds in codimension two. Math. Z. 258, 291–299 (2008)
Florit, L., Hui, W., Zheng, F.: On real Kaehler Euclidean submanifolds with non-negative Ricci curvature. J. Eur. Math. Soc. 7, 1–11 (2005)
Furuhata, H.: Construction and classification of isometric minimal immersions of Kähler manifolds into Euclidean spaces. Bull. Lond. Math. Soc. 26, 487–496 (1994)
Fwu, C.: Kaehler manifolds isometrically immersed in Euclidean space. J. Differ. Geom. 14, 99–103 (1980)
Hasanis, Th.: Isometric immersions of complete Kaehler manifolds into Euclidean spaces. Arch. Math. 38, 470–472 (1982)
Hennes, P.: Weierstrass representations of minimal real Kahler submanifolds. Ph.D. Thesis, State University of New York, Stony Brook (2001)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Interscience Publishers, New York (1969)
Milnor, J.: Morse Theory. Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton (1963)
Moore, J.D., Noronha, M.: Isometric immersions with congruent Gauss maps. Proc. Am. Math. Soc. 110, 463–469 (1990)
Rigoli, M., Tribuzy, R.: The Gauss map for Kählerian submanifolds of \(\mathbb {R}^n\). Trans. Am. Math. Soc. 332, 515–528 (1992)
Ryan, P.: Kähler manifolds as real hypersurfaces. Duke Math. J. 40, 207–213 (1973)
Yan, J., Zheng, F.: A Dajczer-Rodríguez type cylinder theorem for real Kaehler submanifolds. Pure Appl. Math. Q. 9, 563–577 (2013)
Yan, J., Zheng, F.: An extension theorem for real Kaehler submanifolds in codimension four. Mich. Math. J. 62, 421–441 (2013)
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Dajczer, M., Tojeiro, R. (2019). Real Kaehler Submanifolds. In: Submanifold Theory . Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9644-5_15
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