Egor Morozov

Titre : Generic equivariant simplicity of the Laplacian spectrum

Résumé : Let M be a closed smooth manifold. In 1976, K. Uhlenbeck proved that for a generic (in some precise sense) Riemannian metric on M all the eigenvalues of the corresponding Laplace-Beltrami operator are simple (i.e. all the eigenspaces have dimension one). However, if a nontrivial finite group G acts on M and we consider only G-equivariant Riemannian metrics, then the same conclusion does not hold. Indeed, the action of G turns each eigenspace into a G-representation, and it is clear (at least intuitively) that generically not all these representations are trivial (in fact, this follows from an equivariant version of the Weyl law by H. Donnelly). It is then reasonable to expect that for a generic G-equivariant metric all the eigenspaces are not 1-dimensional spaces but irreducible G-representations. In this talk I will describe a new approach to this and related problems based on ideas from symplectic topology. More precisely, it turns out that C. Wendl's proof of the super-rigidity conjecture for J-holomorphic curves can be adapted to this situation. This allows to obtain an equivariant version of Uhlenbeck's result along with few similar results. The talk is based on a joint work with Egor Shelukhin.