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: The domain mathfrak {A} is Kobayashi hyperbolic.The domain mathfrak {A} is Brody hyperbolic.The domain mathfrak {A} possesses a Bergman metric.The domain mathfrak {A} possesses a bounded smooth strictly plurisubharmonic function, i.e. the core mathfrak {c}(mathfrak {A}) of mathfrak {A} is empty.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 rightarrow {mathbb {C}}_w.