The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds

TL;DR

This paper demonstrates that on generic 5-dimensional compact manifolds, the Hodge Laplacian's nonzero eigenvalues on co-exact two-forms typically have multiplicity two, based on spectral properties of the Beltrami operator.

Contribution

It establishes the generic multiplicity of eigenvalues of the Hodge Laplacian on 5-manifolds by linking it to the simple spectrum of the Beltrami operator.

Findings

Eigenvalues of the Hodge Laplacian on co-exact two-forms are generically of multiplicity two.

The spectrum of the Beltrami operator is simple for generic metrics.

The proof connects the Hodge Laplacian to the Beltrami operator via a squared relation.

Abstract

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.