On the existence of Kobayashi and Bergman metrics for Model domains

We prove that for a pseudoconvex domain of the form U = {(z,w) is an element of C-2 : v > F(z, u)}, where w = u + iv and F is a continuous function on C-z x R-u, the following conditions are equivalent: (1) The domain U is Kobayashi hyperbolic. (2) The domain U is Brody hyperbolic. (3) The domain Upossesses a Bergman metric. (4) The domain U possesses a bounded smooth strictly plurisubharmonic function, i.e. the corec (U) of U is empty. (5) The graph Gamma(F) of 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(H) of just one entire function H : C-z -> C-w.