Characterisation of geodesic-preserving functions

Abstract

Let $\Omega_1$, $\Omega_2$ be two domains in $\mathbb{C}^n$ with Kobayashi metrics $k_{\Omega_i}$ and consider a holomorphic mapping $f \in \mathcal{O}(\Omega_1,\Omega_2)$. Let $\mathfrak{F}_1$ and $\mathfrak{F}_2$ be families of geodesics defined on $\Omega_1$ and $\Omega_2$ respectively, where a geodesic between $z$ and $w$ in $\Omega_i$ is the length minimizing curve for the metric $k_{\Omega_i}$. We say that a holomorphic mapping \textit{preserves geodesics} if for any geodesic $\gamma_1$ in $\mathfrak{F}_1$ its image is a subset of a geodesic $\gamma_2$ in $\mathfrak{F}_2$ ($f(\gamma_1)\subset \gamma_2$). We aim to characterise the family of such mappings when $\mathfrak{F}_1$ and $\mathfrak{F}_2$ are the families of Kobayashi geodesics passing through a point in the unit disc $\mathbb{D}$ or in the unit ball $\mathbb{B}^n$. Some additional results are given in the complex plane $\mathbb{C}$ and $\mathbb{C}^n$.