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