On minimal immersions of \(R^ 2\) into \(P^ n(C)\)
type: Item
instance of: scholarly article
title: On minimal immersions of \(R^ 2\) into \(P^ n(C)\)
zbMATH Open document ID: 0563.53048
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: In this paper, complete minimal surfaces in a Kähler manifold are studied. The first theorem of this paper proves that if the square length of the second fundamental form of a compact minimal surface, \(M^ 2\), in \(P^ n(C)\) is smaller than or equal to \(2\rho\), where \(4\rho\) denotes the constant holomorphic sectional curvature of \(P^ n(C)\), then \(M^ 2\) is superminimal or it is flat and totally real in \(P^ n(C)\). Superminimal surfaces in \(P^ n(C)\) have been studied extensively by several mathematicians and physicists (Chern, Eells, Wolfson, Wood, et al.). Therefore we intent to determine all isometric totally real, minimal immersions of the Euclidean 2-plane \(R^ 2\) into \(P^ n(C).\) Main results of this paper show that they constitute an (n-2)-parameter family and each of them must be an orbit of an abelian Lie subgroup of \(U(n+1)\). These results are proved by applying Chern and Wolfson's theory for minimal surfaces in Kähler manifolds.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3897780
zbMATH Keywords: minimal surfaces
zbMATH Keywords: Kähler manifold
zbMATH Keywords: superminimal
zbMATH Keywords: complex projective space
author: Katsuei Kenmotsu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03740665
OpenAlex ID: W1484559311
DOI: 10.2969/JMSJ/03740665
description: scientific article; zbMATH DE number 3897780