  Let $(M^4,g)$ be a closed Riemannian manifold of dimension four. We
investigate the properties of metrics which are critical points of the
eigenvalues of the Paneitz operator when considered as functionals on the space
of Riemannian metrics with fixed volume. We prove that critical metrics of the
aforementioned functional restricted to conformal classes are associated with a
higher-order analog of harmonic maps (known as extrinsic conformal-harmonic
maps) into round spheres. This extends to four-manifolds well-known results on
closed surfaces relating metrics maximizing laplacian eigenvalues in conformal
classes with the existence of harmonic maps into spheres. The case of general
critical points (not restricted to conformal classes) is also studied, and
partial characterization of these is provided.
