The main topic is spectral theory on Riemannian manifolds, and more precisely the study of extremal metrics and of bounds on the spectrum.

A general objective is to choose a metric approach to the problem
and work if possible in the context (or at least in the spirit) of metric measure space. The different aspects are the following:

Riemannian manifolds with an isometric involution. We show that the presence of an isometric involution without fixed point for a compact manifold (M,g) has strong implication for the gap between the two first eigenvalues of Laplace-type operators on a vector bundle over M. In the case of submanifolds or of domains of the Euclidean space, some of these estimates are sharp, and we look at a characterization of equality and almost equality cases.

Qualitative information about extremal eigenvalues.
We want to compare two consecutive extremal eigenvalues for a given problem: typically supremum of the k-th eigenvalue for the Neumann problem of domain of given volume in Euclidean or hyperbolic space.

Laplacian with density. The question of studying the Laplacian with density is interesting in itself, but it appears also in a natural way in the Ph. D. thesis of Th. Barthelmé in relation with the Finsler geometry.


Stability of the spectrum for domains with Dirichlet boundary conditions. We plan to investigate from a more geometric point of view the classical question "if two domains are close, are their Dirichlet spectra also close ?" and to get uniform estimates.

Numerical investigations. A way to have a better understanding of extremal metrics is to make numerical investigations. In the second part of the thesis, we will mainly focus on domains in the hyperbolic plane or in the sphere (for Dirichlet or Neumann boundary conditions), using what was already done in the Euclidean plane.