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

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.