On the existence of Kobayashi and Bergman metrics for Model domains

Abstract

We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( 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 $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

A Kobayashi and Bergman complete domain without bounded representations

“…3 for the definition) possesses a Bergman metric. Some other conditions (for certain unbounded X) which are sufficient for possessing a (complete) Bergman metric are also scattered in the literature, see for examples [2,8,26,28] et al…”

A Kobayashi and Bergman complete domain without bounded representations

“…It was proved there that a pseudoconvex domain with empty core (see Section 3.1 below for the definition) possesses a Bergman metric. Some other conditions (for certain unbounded X) which are sufficient for possessing a (complete) Bergman metric are also scattered in the literatures, see for examples [2], [8], [27], [29] et al…”