  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 $n \geq 2$. As for the Dirichlet Laplacian, we prove non
existence of extremal metrics on any compact surface.
