A note on extremal metrics of non-constant scalar curvature

Abstract

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