Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric
type: Item
instance of: scholarly article
title: Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric
zbMATH Open document ID: 0958.58007
author: Anna Maria Micheletti
author: Angela Pistoia
published in: Discrete and Continuous Dynamical Systems
publication date: 2000-08-16T00:00:00Z
review text: In the past years the dependence of the multiplicity of an eigenvalue on the perturbations of the Riemannian metric has been extensively studied. For instance it has been proved that for generic Riemannian metrics \(g\) the spectrum of \(\Delta g\) consists only of simple eigenvalues. In the paper under review the authors try to get information about the structure of the set of Riemannian metrics which keep fixed the multiplicity of the eigenvalues. More exactly, let \(M\) be a compact, connected \(C^\infty\)-manifold of dimension \(n \geq 2\) without boundary, \(S^k\) the space of all \(C^k\)-symmetric covariant \(2\)-tensors on \(M\) and \(\lambda^\star(g_0)\) a given eigenvalue of the Laplacian \( \Delta_{g_0}\) of multiplicity \(\nu\geq 2\). In this picture they give a sufficient condition such that, in a suitable neighborhood of \(g_0\) in \(S^k\), the set of Riemannian metrics \(g=g_0+h\) for which \(\Delta g\) admits an eigenvalue \(\lambda^\star(g)\) near \(\lambda^\star(g_0)\) of the same multiplicity \(\nu\), is a manifold of \(S^k\) of codimension \(\nu(\nu+1)/2-1\). The case of the 2-torus and 2-spheres are nicely exemplified.
reviewed by: Mircea Puta
Mathematics Subject Classification ID: 58C40
zbMATH DE Number: 1492554
zbMATH Keywords: Laplace operator
zbMATH Keywords: eigenvalue
zbMATH Keywords: multiplicity
zbMATH Keywords: transversality
MaRDI profile type: Publication
full work available at URL: dcds.1998.4.709
OpenAlex ID: W2012758216
DOI: 10.3934/DCDS.1998.4.709
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description: scientific article; zbMATH DE number 1492554