On the Kobayashi hyperbolicity of certain tube domains (Q2845463)

This note is a follow-up on a recent paper by the second author [Geom. Topol. 12, No. 2, 643--711 (2008; Zbl 1143.32016)] in which certain families of tube domains in \(\mathbb C^2\) were introduced as a part of classification of hyperbolic manifolds with \(\text{dim}_{\mathbb C}X=2\) and \(\text{dim}_{\mathbb R} \text{Aut}(X)=3\). Tube domains in \(\mathbb C^2\) have the form \(T_D=D+i\mathbb R^2\) where \(D\) is a domain in \(\mathbb R^2\) called the base of \(T_D\). For example, the family \(T_{C_{\alpha,s,t}}\) is defined in this paper by \(D=\big\{re^{i\varphi}\in \mathbb R^2 : se^{\alpha \varphi}<r<te^{\alpha\varphi} \big\}\) where \(\alpha,t>0\) and \(e^{-2\pi \alpha} t<s<t\).NEWLINENEWLINEA domain \(X\) is Kobayashi hyperbolic if for every point \(x\in X\) there exists a neighborhood \(U\) of \(x\) such that for all holomorphic maps \(f:\{z: |z|<1\}\to X\) with \(f(0)\in U\) the derivative \(df(0)\) is uniformly bounded. This property is equivalent to the Kobayashi pseudodistance being an actual distance. In this note the authors show that all members of the aforementioned families of tube domains are Kobayashi hyperbolic.