A note on extremal metrics of non-constant scalar curvature (Q1802734)

The criterion for a Kähler metric to be extremal and some examples of this type of metrics were given by \textit{E. Calabi} [The space of Kähler metrics, Proc. Int. Congr. Math. Amsterdam 2, 206-207 (1954); Extremal Kähler metrics, in Semin. differential geometry, Ann. Math. Stud. 102, 259-290 (1982; Zbl 0487.53057)]. The main purpose of this paper is to obtain extremal Kähler metrics of non-constant scalar curvature on the blow-up of \(\mathbb{C}^ n\) at \(\vec 0\), by working in \(\mathbb{C}^ n\) with potentials of the form \(a\log+ s(u)\), \(u\) the square of the distant to the origin. The author reobtains Calabi's metrics by the completion at \(\infty\) of these metrics adding \(\mathbb{C} \mathbb{P}^{n-1}\). This approach illuminates further Calabi's construction, as some of the technicalities of his construction are eliminated. This type of metrics is constructed also on other bundles over \(\mathbb{C} \mathbb{P}^{n-1}\).