Bergman completeness of unbounded Hartogs domains. (English) Zbl 1094.32004
The authors study the Bergman metric of unbounded pseudoconvex domains in \(\mathbb C^n\). After generalizing a few results up to now known for bounded domains only to the unbounded case, they present a new class of Bergman complete not hyperconvex unbounded domains: Hartogs domains \(D_\rho=\{(z,z_{n+1})\in\mathbb C^{n+1}\mid | z_{n+1}| <e^{-\rho(z)}\}\), where \(\rho:\mathbb C^n\to\mathbb R\) is a plurisubharmonic function bounded from below and such that \(\lim_{\| z\| \to\infty}\| z\| ^k e^{-\rho(z)}=0\) for all \(k>0\).
Reviewer: Marco Abate (Pisa)
MSC:
32F45 | Invariant metrics and pseudodistances in several complex variables |
32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |
30C85 | Capacity and harmonic measure in the complex plane |
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