We consider an eigenvalue problem for the biharmonic operator on bounded domains of Riemannian manifolds with Neumann boundary conditions. We will discuss classical boundary conditions as well as the weak formulation of the problem. Then, we will present a few properties of the eigenvalues. In particular, we will discuss upper bounds which are compatible with the Weyl's law. Also, we shall provide examples where, differently from the corresponding biharmonic Dirichlet problem (on manifolds) or the analogous biharmonic Neumann problem on Euclidean domains, there exist negative eigenvalues. If time permits, we shall present a few more examples describing some peculiar behavior of the eigenvalues, as well as a number of open questions.
Based on a joint work with Bruno Colbois (Université de Neuchâtel).