Abstract: This thesis explores parametric arguments in the spectral theory of the curl and Laplace-Beltrami operator, motivated by problems coming from contact geometry and topology.
Contact topology is the study of the topology of totally non-integrable hyperplane fields, and its importance to low dimensional topology is well known. A characterization of contact forms in dimension 3 as nowhere vanishing solutions to the eigenvalue problem of the curl operator opens up a geometric-analytic approach to the subject. One of the aims of this thesis is to build the spectral theoretic foundation for future investigations of this research direction.
To this end, we prove generic spectral simplicity of the curl operator along 1-parameter families of Riemannian metrics, a result we hope will lead to flexibility results for eigenforms of the curl operator. This project is motivated by the fundamental importance 1-parameter families of contact structures in contact topology.
In joint work with Josef Greilhuber, far reaching generalizations of these arguments are used to resolve a conjecture of Vladimir Arnold on the splitting behaviour of multiple eigenvalues of the Laplace-Beltrami operator under perturbation of the metric. Due to this result we now know the codimension of the space of metrics for which the Laplace-Beltrami operator has at least one eigenvalue of multiplicity m.
The fact that contact forms in dimension three are eigenforms of the curl operator and present a rich class of stationary solutions to the three dimensional Euler equations can be exploited to construct exotic solutions to those equations. Combining input from dynamical systems with a good understanding of the perturbation theory of the curl operator allows one to engineer a stationary solution to the three dimensional Euler equations which is isolated in the C 1 topology. This part of the thesis is joint work with Alberto Enciso and Daniel Peralta-Salas.
Another result detailed in this thesis, again obtained in collaboration with Josef Greilhuber, concerns the spectral geometry of the curl operator on smoothly bounded domains in three dimensional Euclidean space. We show that for generic smoothly bounded domains, the spectrum of the curl operator consists of discrete eigenvalues of multiplicity one. As a crucial step we derive a variational formula for the first-order change of eigenvalues of the curl operator along deformations of the underlying smoothly bounded domain.

univienna.zoom.us/j/65382614762
Meeting ID: 653 8261 4762
Kenncode: 710118