Abstract: Extremal Kaehler metrics of non constant scalar curvature
have been constructed by Calabi on all Hirzebruch sufaces and in all
Kaehler classes. . His method easily extends to provide extremal
Kaehler metrics on Hirzebruch-like ruled surfaces of any genus, but
only on a part of the Kaehler cone when the genus is greater than 1.
A recent theorem by X. X.Chen and G. Tian allows to conclude to the non-existence of extremal metrics for these Kaehler classes. The argument relies on an explicit calculation of the (modified) Mabuchi energy.