Sur la multiplicit\'e des valeurs propres du laplacien de Witten

Abstract

On any compact manifold of dimension greater than 4, we prescribe the volume and any finite part of the spectrum of the Witten Laplacian acting on $p$-form for $0<p<n$. In particular, we prescribe the multiplicity of the first eigenvalues. On 3-dimensional manifolds, we give examples of multiple first eigenvalue for 1-forms, whose multiplicity depends on the maximal genus of embedded surfaces all of whose 1-cohomology is induced by the cohomology of the manifold. In particular, this multiplicity is at least 3.