On the existence of Kobayashi and Bergman metrics for Model domains
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Mathematische Annalen
On the existence of Kobayashi and Bergman metrics for Model domains Nikolay Shcherbina1 Received: 22 October 2019 / Revised: 21 July 2020 © The Author(s) 2020
Abstract We prove that for a pseudoconvex domain of the form A = {(z, w) ∈ C2 : v > F(z, u)}, where w = u + iv and F is a continuous function on Cz × Ru , the following conditions are equivalent: (1) The domain A is Kobayashi hyperbolic. (2) The domain A is Brody hyperbolic. (3) The domain A possesses a Bergman metric. (4) The domain A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core c(A) of A is empty. (5) The graph (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 (H) of just one entire function H : Cz → Cw . Mathematics Subject Classification Primary 32T99 · 32F45 · 32U05; Secondary 32Q45
Contents 1 2 3 4 5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ . . . . . . . . . . . . . . . . . . . . . . . . . Construction of the domain A ˜ =A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case when A ˜ = A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case when A 5.1 Construction of the limit disk passing through the given point of F . . . 5.2 Construction and properties of the maximal limit leaf L0 contained in E0 5.3 Foliation structure of ∂A . . . . . . . . . . . . . . . . . . . . . . . . . 6 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Communicated by Ngaiming Mok.
B 1
Nikolay Shcherbina [email protected] Department of Mathematics, University of Wuppertal, 42119 Wuppertal, Germany
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N. Shcherbina
1 Introduction The purpose of this paper is to study the Kobayashi and Bergman metrics for pseudoconvex domains of the form A = {(z, w) ∈ Cnz × Cw : v > F(z, u)}, where w = u + iv and F is a continuous function on Cnz × Ru . This type of domains we call in what follows for Model domains. They appear naturally (usually with much more special choice of the function F) as the limit domains in the scaling method (see, for example, [23]), in the representations of domains of finite type, in the biholomorphic classification problem etc. Note that in the last problem the existence of Kobayashi metric on A is of special interest, since in the case of its existence the group Aut(A) of holomorphic automorphisms of A is in fact a finite-dimensional real Lie group (see, for example, [19, Folgerung 2.6, p. 55] or [20, Theorem 2.1, p. 68]). Despite the fact that many properties
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