2022/11/13 by Willi Kepplinger, Kepplinger, Willi
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2211.06916
openalex publication_date 2022/11/13 · openalex created_date 2022/11/23 · openalex updated_date 2026/07/28
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 Ck Riemannian metrics, where k≥ 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 \citeTeytel1999, 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 Ck topology as proven by Enciso and Peralta-Salas in \citeEnciso2012, it is not a meagre codimension 2 property.