Spectral geometry of the second variation operator of harmonic maps
type: Item
instance of: scholarly article
title: Spectral geometry of the second variation operator of harmonic maps
zbMATH Open document ID: 0644.53048
author: Hajime Urakawa
published in: Illinois Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The inverse spectral problem of the Hessian, i.e. the Jacobi operator \(J_{\phi}\) of the energy of a harmonic map \(\phi\) is dealt with. For the spectral geometry for the Laplace-Beltrami operator of a compact Riemannian manifold, several interesting geometric results are obtained. Analogously, using the data of the spectrum \(Spec(J_{\phi})\) of \(J_{\phi}\) of a harmonic map \(\phi\), typical harmonic maps, i.e., constant maps, geodesics, isometric minimal immersions, harmonic Riemannian submersions are characterized. For example, let \(\phi\), \(\phi\) ' be harmonic maps of (S 2,can) into the complex projective space (P n(\({\mathbb{C}}),h)\) with the Fubini-Study metric h. Assume that \(Spec(J_{\phi})=Spec(J_{\phi '})\). If \(\phi\) is a holomorphic isometric immersion, then \(\phi '=\phi\) up to an isometry of (P n(\({\mathbb{C}}),h)\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 4050478
zbMATH Keywords: Jacobi operator
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: inverse spectral problem
zbMATH Keywords: Hessian
zbMATH Keywords: harmonic map
zbMATH Keywords: geodesics
zbMATH Keywords: minimal immersions
zbMATH Keywords: harmonic Riemannian submersions
zbMATH Keywords: holomorphic isometric immersion
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4050478