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

Abstract

For a domain $D\subset{\Bbb C}$ the Kobayashi--Royden $\kappa$ and Hahn $h$ pseudometrics are equal iff $D$ is simply connected. Overholt showed that for $D\subset{\Bbb C}^n$, $n\geq3$, we have $h_D\equiv\kappa_D$. Let $D_1,D_2\subset{\Bbb C}$. 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 ${\Bbb C}\setminus\{0\}$. In particular, there are domains $D\subset{\Bbb C}^2$ for which $h_D\not\equiv\kappa_D$.