## 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.)

### Keywords:

Laplace operator; spectrum; first eigenvalue
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