# On the existence of Kobayashi and Bergman metrics for Model domains

## TL;DR

This paper establishes the equivalence of several complex geometric properties, including hyperbolicity and the existence of Bergman metrics, for a class of pseudoconvex model domains in complex two-space.

## Contribution

It proves that for certain pseudoconvex domains, hyperbolicity, Bergman metric existence, and plurisubharmonic functions are equivalent, characterizing their complex geometry.

## Key findings

- Kobayashi hyperbolicity implies Bergman metric existence.
- Absence of special holomorphic foliation characterizes hyperbolicity.
- Equivalence of hyperbolicity, Bergman metric, and plurisubharmonic core.

## 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 ${\mathcal H} : {\mathbb C}_z \to {\mathbb C}_w$.