Minimal immersions of Kähler manifolds into complex space forms
type: Item
instance of: scholarly article
title: Minimal immersions of Kähler manifolds into complex space forms
zbMATH Open document ID: 0627.53047
author: Seiichi Udagawa
published in: Tokyo Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: As it is well known, holomorphic isometric immersions of Kähler manifolds into Kähler manifolds are minimal immersions. But minimal isometric immersions of Kähler manifolds are not always holomorphic or anti-holomorphic. In the present paper, the author obtains the following results: Theorem 1. Let \(CH^ m(c)\) be an m-dimensional complex hyperbolic space of constant holomorphic sectional curvature c \((c<0)\), and let M be an n- dimensional Kähler manifold such that dim \(M_ c=n\geq 2\). Then, every minimal isometric immersion of M into \(CH^ m(c)\) is a holomorphic or anti-holomorphic immersion. Theorem 2. Let \(f: M\to R^{2n+2}/D\) be an isometric stable minimal immersion of an n-dimensional compact Kähler manifold into a \((2n+2)\)-dimensional flat torus. Assume that \(| R|^ 2\geq \tau^ 2\) holds on M, where R is the curvature tensor and \(\tau\) is the scalar curvature of M. Then, f is holomorphic with respect to some orthogonal complex structure of \(R^{2n+2}/D\). The last result is a generalization of \textit{M. J. Micallef} [J. Differ. Geom. 19, 57-84 (1984; Zbl 0537.32010)].
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 4019945
zbMATH Keywords: complex space form
zbMATH Keywords: minimal immersions
zbMATH Keywords: holomorphic immersion
zbMATH Keywords: stable minimal immersion
zbMATH Keywords: Kähler manifold
reviewed by: Toru Ishihara
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270141806
OpenAlex ID: W2086010936
DOI: 10.3836/TJM/1270141806
Recommended article: Holomorphicity of minimal submanifolds in complex space forms
Recommended article: Kaehler submanifolds of hyperbolic space
Recommended article: On isometric immersions into complex space forms
Recommended article: Construction and Classification of Isometric Minimal Immersions of Kähler Manifolds into Euclidean Spaces
Recommended article: MINIMAL IMMERSIONS OF KÄHLER MANIFOLDS INTO EUCLIDEAN SPACES
Recommended article: Q4892954
Recommended article: Holomorphicity of real Kaehler submanifolds
Recommended article: Q4392797
Recommended article: Q4298768
Recommended article: Q5201886
description: scientific article; zbMATH DE number 4019945