Extremal metrics for the eigenvalues of the Laplacian on manifolds with boundary

We show there are no extremal metrics for the eigenvalues of the Neumann Laplacian on any compact manifold. Nonetheless, we construct examples of conformally extremal metrics for the eigenvalues of this operator in any annulus and characterise these special metrics in the general case of a compact manifold of dimension $n\ge 2$. As for the Dirichlet Laplacian, we prove non existence of extremal metrics on any compact surface.