A note on extremal metrics of non-constant scalar curvature

Abstract

By working in ℂ n with potentials of the forma logu + s(u), u the square of the distant to the origin, we obtain extremal Kahler metrics of nonconstant scalar curvature on the blow-up of ℂ n at $$\vec 0$$ . We then show that these metrics can be completed at ∞ by adding a ℂℙ n−1, and reobtain the extremal Kahler metrics of non-constant scalar curvature constructed by Calabi on the blow-up of ℂℙ n at one point. A similar construction produces this type of metrics on other bundles over ℂℙ n − 1.