On the existence of Kobayashi and Bergman metrics for Model domains

2019/04/29 by Shcherbina, Nikolay
#32F45 #32T99 #32U05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.12950

Abstract

We prove that for a pseudoconvex domain of the form \mathfrakA = \(z, w) ∈ \mathbb C² : v > F(z, u)\, where w = u + iv and F is a continuous function on \mathbb C_z × \mathbb R_u, the following conditions are equivalent: (1) The domain \mathfrakA is Kobayashi hyperbolic. (2) The domain \mathfrakA is Brody hyperbolic. (3) The domain \mathfrakA possesses a Bergman metric. (4) The domain \mathfrakA possesses a bounded smooth strictly plurisubharmonic function, i.e. the core \mathfrakc(\mathfrakA) of \mathfrakA is empty. (5) The graph 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 Γ(\mathcal H) of just one entire function \mathcal H : \mathbb C_z → \mathbb C_w.

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