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

TL;DR

This paper demonstrates the ability to prescribe the spectrum and eigenvalue multiplicities of the Witten Laplacian on compact manifolds, providing explicit examples and conditions for various dimensions.

Contribution

It establishes new results on prescribing the spectrum and multiplicities of the Witten Laplacian on manifolds of different dimensions, including explicit constructions in 3D.

Findings

Prescribable spectrum and eigenvalue multiplicities on higher-dimensional manifolds.

Existence of multiple first eigenvalues on 3D manifolds with genus-dependent multiplicity.

At least triple multiplicity for certain 1-form eigenvalues in 3D cases.

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.