In this talk, we will discuss some partial results in a project initiated by Vestislav Apostolov aiming to solve a conjecture about the specific form of an extremal Kahler metric on a projective bundle over a curve. The conjecture, on one hand, connects to the Donaldson-Uhlenbeck-Yau Theorem and, on the other hand relates the existence of extremal metrics to the stability of the vector bundle, which is a natural extension of Yau's conjecture in the case of Kaehler Einstein metric when the first Chern class of the Kaehler manifold is positive.