Extremal metrics on ruled manifolds

In this paper, we consider a compact Kähler manifold with extremal Kähler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kähler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kähler metrics on the projectivization of the dual vector bundle.