Seminario Geometría

On the Simplicity of the Spectrum of the Hodge Laplacian along Paths of Riemannian Metrics

Ponente:  Willi Kepplinger (University of Vienna )
Fecha:  jueves 12 de enero de 2023 - 11:30
Lugar:  Aula Naranja, ICMAT


When given a family of operators dependent on some parameter space, it is often of interest to ask for generic properties of these operators, i.e. properties that hold for a generic subset of the parameter space. In particular generic properties of eigenfunctions and eigenvalues have generated substantial interest, starting with a landmark paper by Uhlenbeck (Generic Properties of Eigenfunctions, American Journal of Mathematics (1976)) in which she proves, among other things, that the spectrum of the Laplace Beltrami operator is generically simple on closed Riemannian manifolds(here the parameter space is the space of Riemannian metrics), even along generic paths of Riemannian metrics. Several attempts have been made to determine to which extend these results generalize to other geometric operators. We will present our recent article in which we show that while the Hodge Laplacian on closed oriented 3-manifolds does not have simple spectrum along generic paths of Riemannian metrics, we understand why these higher multiplicities arise and retrieve simplicity along generic paths for the closed, positive and negative coclosed spectrum of the Hodge Laplacian in dimension 3.