Extremal Kähler Metrics on Ruled Surfaces

We will define a special type of Riemannian metric, called an admissible Kähler metric, on a certain type of 4-dimensional manifold. A metric of this type is determined by a positive and smooth function F(z) which satisfies certain boundary conditions. When this F (z) is also an extremal point of a certain functional, the scalar curvature of the metric is a linear function of z. Metrics determined by such functions are called "admissible extremal metrics," which are special cases of Calabi's extremal metrics. We will then discuss the existence of such metrics on our type of manifolds.