On spectral simplicity of the Hodge Laplacian and Curl Operator along paths of metrics

Abstract

We prove that the curl operator on closed oriented $3$-manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has $1$-dimensional eigenspaces, even along $1$-parameter families of $\mathcal{C}^k$ Riemannian metrics, where $k\geq 2$. We show further that the Hodge Laplacian in dimension $3$ has two possible sources for nonsimple eigenspaces along generic $1$-parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel \cite{Teytel1999}, allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension $3$ is a meagre codimension $1$ property with respect to the $\mathcal{C}^k$ topology as proven by Enciso and Peralta-Salas in \cite{Enciso2012}, it is not a meagre codimension $2$ property.