Extremal Kähler metrics on unstable projectivised vector bundles

We investigate the problem of constructing extremal Kähler metrics on the projectivisation of an unstable vector bundle (unstable in the sense of Mumford-Takemoto). This provides us with new examples of extremal Kähler metrics not having constant scalar curvature. The main technique employed to prove the existence results is an adiabatic limit, as already used in earlier works by Hong and Fine. I will talk about the general ideas and an outline of the existence proof, without worrying too much about the analytical and technical details.