Multiple eigenvalues of the Laplace-Beltrami operator and deformation of the Riemannian metric
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.
scientific article; zbMATH DE number 1492554