Kobayashi-Royden vs. Hahn pseudometric in ${\mathbb{C}}^2$

Abstract

For a domain $D\subset \mathbb{C}$ the Kobayashi--Royden $\kappa$ and Hahn $h$ pseudometrics are equal iff $D$ is simply connected. Overholt showed that for $D\subset {\mathbb{C}}^n$, $n\ge 3$, we have $h_D\equiv {\kappa }_D$. Let $D_1,D_2\subset \mathbb{C}$. The aim of this paper is to show that $h_{D_1\times D_2}\equiv {\kappa }_{D_1\times D_2}$ iff at least one of $D_1$, $D_2$ is simply connected or biholomorphic to $\mathbb{C}\setminus \{0\}$. In particular, there are domains $D\subset {\mathbb{C}}^2$ for which $h_D\not\equiv {\kappa }_D$.

5 pages