Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds

TL;DR

This paper proves that for generic metrics on closed 3-manifolds, the nonzero eigenvalues of the Hodge Laplacian are simple and the zero sets of eigenforms are isolated, using analysis of the Beltrami operator.

Contribution

It establishes generic simplicity of eigenvalues and isolated zero sets of eigenforms for the Hodge Laplacian on 3-manifolds, extending understanding of spectral properties.

Findings

Nonzero eigenvalues are simple for generic metrics.

Zero sets of eigenforms are isolated points.

Results hold for residual sets of $C^r$ metrics with $r \\geq 2$.

Abstract

In this paper we analyze the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold $M$. In particular, we show that the nonzero eigenvalues are simple and the zero set of the eigenforms of degree 1 or 2 consists of isolated points for a residual set of $C^r$ metrics on $M$, for any integer $r\geq2$. The proof of this result hinges on a detailed study of the Beltrami (or rotational) operator on co-exact 1-forms.