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 3 -manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1 1 -dimensional eigenspaces, even along 1 1 -parameter families of C k \mathcal {C}^k Riemannian metrics, where k ≥ 2 k\geq 2 . We show further that the Hodge Laplacian in dimension 3 3 has two possible sources for nonsimple eigenspaces along generic 1 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 [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], 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 3 is a meagre codimension 1 1 property with respect to the C k \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 2 property.