A note on higher extremal metrics

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.