Geometric Spectral theory - (:unkn) unknown (My University, 2025)
The main topic of this proposal is spectral geometry on Riemannian manifolds, that is the relationship between the geometry (or the topology) of a Riemannian manifold, submanifold or domain, and the spectrum of a Laplace-type operator. The Laplacian on a Riemannian manifold and the study of the relationship between the spectrum of the Laplacian and the geometry/topology of the manifold are now classical and well-studied subjects. However, for other Laplace-type (or related) operators, the relationship between the spectrum and the geometry or topology of the underlying manifold is much less clear, and the aim of this proposal is to investigate this relationship for three operators: the Steklov (or Dirichlet-to-Neumann) operator, the magnetic Laplacian and the bi-Laplacian. In recent years, the Steklov eigenvalue problem has been extensively studied, especially from a geometric point of view. It is defined on a manifold with boundary, and what we want to understand better is how the geometry of the boundary and the geometry of the whole manifold interact in order to influence the Steklov spectrum. The investigation of the spectrum of the magnetic Laplacian is a continuation of a previous collaboration. We would like to understand a family of ``simple'' examples, like Euclidean domains of dimension 2. In such situations, the spectrum (and in particular the first eigenvalue) is very sensitive to the topology of the domain. Regarding the bi-Laplacian, our goal is to investigate the bi-laplacian with Neumann boundary conditions on compact Riemannian manifolds with boundary. Our recent substantial results lead to a number of questions that we want to investigate from a geometric viewpoint, which is a totally new approach for this type of operator. For example, how to find general upper bounds for the spectrum under the condition that the Ricci curvature is bounded from below? Another direction is the study of domains in hyperbolic space or in Cartan-Hadamard manifolds, where negative eigenvalues appear. Initially, my research topics were often studied from an analytical point of view: this is due to the fact that they were (and still are) dealt with in Euclidean space where powerful analytical methods are available. The common theme of all my research work is that I seek to understand the different problems that interest me from the geometric and metric points of view. On the one hand, this allows me to deal with them in a more general framework, but also to have a point of view that differs from that of an analytical approach. In particular, the metric point of view allows to us to get rather robust estimates regarding variations of the Riemannian metric.
Keywords: Spectral theory on Riemannian manifolds; Upper and lower bounds on the spectrum; Laplace-type operator; Magnetic Laplacian; Steklov operator; bi-Laplacian