Colin de Verdière, Yves Sur la multiplicité de la première valeur propre non nulle du Laplacien. (On the multiplicity of the first nonzero eigenvalue of the Laplacian). (French) Zbl 0607.53028 Comment. Math. Helv. 61, 254-270 (1986). Let (M,g) be a compact Riemannian manifold. Let \(\Delta\) be the Laplace operator for the vector \(C^{\infty}(M)\) of all the functions on the manifold M. The spectrum of \(\Delta\) has the form \(S_ p(M,g)=\{0<\lambda_ 1=...=\lambda_ 1<\lambda_ 2...\lambda_ 2<...<\infty \}\). One of the problems of the spectrum is to study some properties of the first eigenvalue of the \(S_ p(M,g)\). The main result of this paper can be stated as follows. Let M be a compact differentiable manifold of dimension \(n\geq 3\). If N is an arbitrary integer, then there is a metric g on M such that the first eigenvalue of \(S_ p(M,g)\) has multiplicity N. Reviewer: G.Tsagas Cited in 3 ReviewsCited in 33 Documents MSC: 53C20 Global Riemannian geometry, including pinching 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) Keywords:Laplace operator; spectrum; first eigenvalue PDFBibTeX XMLCite \textit{Y. Colin de Verdière}, Comment. Math. Helv. 61, 254--270 (1986; Zbl 0607.53028) Full Text: DOI EuDML