Stability of harmonic maps and standard minimal immersions
type: Item
instance of: scholarly article
title: Stability of harmonic maps and standard minimal immersions
zbMATH Open document ID: 0585.58012
author: Yoshihiro Ohnita
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1986-01-01T00:00:00Z
review text: By the results of Xin and Leung the standard n-sphere \(S^ n\) with \(n\geq 3\) has the property that (1) a noncontant harmonic map from \(S^ n\) to any Riemannian manifold is unstable, and (2) a nonconstant harmonic map from any compact Riemannian manifold to \(S^ n\) is unstable. If a compact Riemannian manifold has this property, then we call it harmonically unstable. In this note we classify harmonically unstable symmetric spaces. We show that a compact symmetric space M is harmonically unstable, if and only if M is a product of simply connected compact irreducible symmetric spaces belonging to the following list: (i) SU(n) (n\(\geq 2)\), Sp(n) (n\(\geq 2)\), (ii) SU(2n)/Sp(n) (n\(\geq 3)\), (iii) \(S^ n\) (n\(\geq 3)\), (iv) \(Sp(p+q)/Sp(p)\times Sp(q)\) (p\(\geq q\geq 1)\), (v) \(E_ 6/F_ 4\), (vi) \(F_ 4/Spin(9)\). This list is the complete list of compact irreducible symmetric spaces with the unstable identity map as a harmonic map. Furthermore we study the harmonic instability of isoparametric hypersurfaces in a unit sphere.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 3938078
zbMATH Keywords: stability of harmonic maps
zbMATH Keywords: standard minimal immersions
zbMATH Keywords: compact irreducible symmetric spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228492
OpenAlex ID: W2010747355
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Wikidata QID: Q125973907
DOI: 10.2748/TMJ/1178228492
description: scientific article; zbMATH DE number 3938078