  In this paper we introduce higher extremal Kahler metrics. We provide an
example of the same on a minimal ruled surface. We also prove a perturbation
result that implies that there are non-trivial examples of higher constant
scalar curvature metrics, which are basically metrics where the top Chern form
is harmonic. We also give a relatively short proof of a formula due to Liu for
the Bando-Futaki invariants (which are obstructions for the existence of
harmonic Chern forms) of hypersurfaces in projective space.
