Holomorphicity of minimal submanifolds in complex space forms
type: Item
instance of: scholarly article
title: Holomorphicity of minimal submanifolds in complex space forms
zbMATH Open document ID: 0631.53046
DOI: 10.1007/BF01458320
author: Marcos Dajczer
author: Gudlaugur Thorbergsson
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164254
review text: 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.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 32C15
zbMATH DE Number: 4026293
zbMATH Keywords: Kählerian manifolds
zbMATH Keywords: holomorphicity
zbMATH Keywords: minimal immersions
zbMATH Keywords: complex space forms
reviewed by: Fritz Gackstatter
MaRDI profile type: Publication
cites work: Some remarks on a class of submanifolds in space forms of non-negative curvature
cites work: Minimal Surfaces by Moving Frames
cites work: A Characterization of Complex Hypersurfaces in C m
cites work: Real Kaehler submanifolds and uniqueness of the Gauss map
cites work: Rigidity of real Kaehler submanifolds
cites work: The fundamental equations of minimal surfaces in \({\mathbb{C}}P^ 2\)
cites work: Symmetric submanifolds of Euclidean space
cites work: Parallel submanifolds of complex space forms I
cites work: Q3721461
cites work: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds
cites work: Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems
cites work: Minimum imbeddings of compact symmetric spaces of rank one
cites work: Minimal immersions of Kähler manifolds into complex space forms
description: scientific article; zbMATH DE number 4026293