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 $k\ge 2$. 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.

There was a gap in the proofs of Lemmas 3.7 and 3.8 which in principle can be fixed by careful perturbation theory, but we have opted to instead rely on an explicit deformation of the round sphere due to S. Tanno and subsumed the core content of both Lemmas in what is now called Proposition 3.7. The corresponding erratum has been accepted by TAMS