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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

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

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.)
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