Geometric Spectral Theory
Responsable du projet Bruno Colbois
Collaborateur Luc Pétiard
Résumé The main topic of this proposal is
"spectral geometry on Riemannian manifolds", and more precisely the
study of lower and upper bounds for the spectrum of the Laplacian and of generalized Laplacians or analogous operators.

The classical Laplacian may be generalized in many directions, and during these last years, there were in particular a lot of investigations around the Laplacian on a weighted manifold, the Steklov (or Dirichlet-to-Neumann) operator and the magnetic Laplacian. In this proposal, I focus on a geometrical approach of the spectrum: I am mainly interested in finding uniform bounds which are of metric nature (a typical example being isoperimetric constants). Such are rather robust regarding variations of the Riemannian metric. I am also looking for uniform upper bounds as compatible as possible with the Weyl asymptotic formula.

In this context, I am asking for two fellowships, one for a doctoral student, Luc Pétiard, and one for a post-doc, Guillaume Roy-Fortin.

In the following, I present separately different projects, even if there will certainly be interactions between them.

With A. El Soufi, we will mainly investigate the Laplacian in a manifold with densities: we will study the problem $-\mbox{div}(\sigma \nabla u)=\lambda \rho u $ in the conformal class [g_0] of a Riemannian manifold (M,g_0), which was already adressed in [CES]. We will study the interaction between $\rho,\sigma$ and $[g_0]$, try to find good uniform upper bounds in special cases and a Cheeger type lower bound for the first nonzero eigenvalue. We will discuss the sharpness of the upper bounds and try to find extremal manifolds/domains in some special situations (first nonzero eigenvalue).
This will also be the context of the Ph.D. thesis of L. Pétiard.

In collaboration with A. Girouard, the aim is to investigate the spectrum of the Steklov operator. There are two main topics, the first being the rough discretization of the spectrum and application (also in collaboration with B. Raveendran). The second will be an investigation of the Steklov spectrum for a family of Riemannian manifolds with fixed boundary, in order to measure how the restriction of fixing the boundary implies some rigidity.

A more prospective part of the project will be to investigate the magnetic Laplacian in collaboration with A. Savo. There has been a lot of interest around the magnetic Laplacian in relation with physics. We are interested in a geometrical approach taking into account the geometry and the topology of the underlying manifold, which seems to be rather new in this context.

The collaboration with G. Roy-Fortin will focuss on the shape optimization for the Neumann problem and the study of generalized laplacian, such as the bi-Laplacian.
Mots-clés spectral theory on Riemannian manifolds, upper and lower bound on the spectrum,
Laplace-type operators, Steklov operator, magnetic Laplacian, extremal metrics
Type de projet Recherche fondamentale
Domaine de recherche Mathematics (Riemannian geometry)
Source de financement FNS - Encouragement de projets (Div. I-III)
Etat Terminé
Début de projet 1-10-2015
Fin du projet 30-9-2018
Budget alloué 313'522.00
Contact Bruno Colbois