Dajczer, Marcos; Thorbergsson, Gudlaugur Holomorphicity of minimal submanifolds in complex space forms. (English) Zbl 0631.53046 Math. Ann. 277, 353-360 (1987). Let M be a Kähler manifold and let \({\mathbb{C}}Q_ c\) be a complex space form with holomorphic sectional curvature c. The problem of the present paper is to find conditions which imply that a minimal isometric immersion f of M into \({\mathbb{C}}Q_ c\) is holomorphic. The authors give a complete answer to the problem in the case \(c<0:\) If \(\dim_{{\mathbb{C}}} M>1\), then f is holomorphic or antiholomorphic, if \(\dim_{{\mathbb{C}}} M=1\), then there are examples of minimal immersions which are not holomorphic. It is not enough to restrict the dimension of M if \(c>0\), but if f is a circular immersion a similar theorem holds for positively curved complex space forms. Reviewer: F.Gackstatter Cited in 6 Documents MSC: 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 32C15 Complex spaces Keywords:Kählerian manifolds; holomorphicity; minimal immersions; complex space forms × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Abe, K.: Some remarks on a class of submanifolds in space forms of non-negative curvature. Math. Ann.247, 275-278 (1980) · Zbl 0439.53058 · doi:10.1007/BF01348959 [2] Chern, S.S, Wolfson, J.G.: Minimal surfaces by moving frames. Am. J. Math.105, 59-83 (1983) · Zbl 0521.53050 · doi:10.2307/2374381 [3] Dajczer, M.: A characterization of complex hypersurfaces inC m . Preprint · Zbl 0661.53014 [4] Dajczer, M., Gromoll, D.: Real Kaehler submanifolds and uniqueness of the Gauss map. J. Differ. Geom.22, 13-28 (1985) · Zbl 0587.53051 [5] 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] Eschenburg, J.H., Guadalupe, I.V., Tribuzy, R.: The fundamental equations of minimal surfaces inCP 2. Math. Ann.270, 571-598 (1985) · Zbl 0536.53056 · doi:10.1007/BF01455305 [7] Ferus, D.: Symmetric submanifolds of Euclidean space. Math. Ann.247, 81-93 (1980) · Zbl 0446.53041 · doi:10.1007/BF01359868 [8] Naitoh, H.: Parallel submanifolds of complex space forms. I. Nagoya Math. J.90, 85-117 (1983) · Zbl 0509.53046 [9] Rawnsley, J.:f-structures,f-twistor spaces and harmonic maps. In: Lect. Notes Math. 1164. Berlin, Heidelberg, New York: Springer 1986 · Zbl 0592.58009 [10] Siu, Y.T.: The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. Math.112, 73-111 (1981) · Zbl 0517.53058 · doi:10.2307/1971321 [11] Siu, Y.T.: Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems. J. Differ. Geom.17, 55-138 (1982) · Zbl 0497.32025 [12] Tai, S.S.: Minimum imbeddings of compact symmetric spaces of rank one. J. Differ. Geom.2, 55-66 (1968) · Zbl 0164.22302 [13] Udagawa, S.: Minimal immersions of Kähler manifolds into complex space forms. Preprint · Zbl 0627.53047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.