On Grauert's examples of complete K\"{a}hler metrics

Abstract

Grauert showed that the existence of a complete K\"{a}hler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb{C}^n \setminus \{0\}, n \ge 2$ and $\mathbb{B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb{B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of its restriction to the leaves of a suitable holomorphic foliation of these two examples. We also examine this metric on the punctured plane $\mathbb{C}^{\ast}$ and show that it behaves very differently in this case.