In the first part of this thesis we investigate the deformation theory of
compact constant scalar curvature Kähler (cscK) and extremal Kähler manifolds.
In particular, we will characterise deformations of cscK- and extremal
Kähler manifolds which again admit a cscK- or extremal Kähler metric, in
terms of stability in the sense of geometric invariant theory.
In a second part we will investigate a different problem. We will extend
the adiabatic methods used by Hong and Fine in the study of cscK-metrics
to the study of extremal metrics on ruled manifolds. The analysis in the case
of extremal metrics being more involved will bring in some new aspects in
carrying out this construction
