On the existence of Kobayashi and Bergman metrics for Model domains

Abstract

We prove that for a pseudoconvex domain of the form $\mathfrak{A}=\{(z,w)\in {\mathbb{C}}^2:v>F(z,u)\}$, where $w=u+iv$ and F is a continuous function on ${\mathbb{C}}_z\times {\mathbb{R}}_u$, the following conditions are equivalent: (1) The domain $\mathfrak{A}$ is Kobayashi hyperbolic. (2) The domain $\mathfrak{A}$ is Brody hyperbolic. (3) The domain $\mathfrak{A}$ possesses a Bergman metric. (4) The domain $\mathfrak{A}$ possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $\mathfrak{c}(\mathfrak{A})$ of $\mathfrak{A}$ is empty. (5) The graph $\Gamma (F)$ of $F$ can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $\Gamma (\mathcal{H})$ of just one entire function $\mathcal{H}:{\mathbb{C}}_z\to {\mathbb{C}}_w$.

24 pages. Comments welcome