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 |
References:
| [1] | Diss. Math. 388 pp 1– (2000) |
| [2] | Bull. Pol. Acad.: Math. 50 pp 297– (2002) |
| [3] | (1995) |
| [4] | DOI: 10.1007/s0013-003-0096-6 · Zbl 1037.30009 · doi:10.1007/s0013-003-0096-6 |
| [5] | DOI: 10.1007/BF01166457 · Zbl 0625.32011 · doi:10.1007/BF01166457 |
| [6] | Walter de Gruyter 8 (1993) |
| [7] | Nagoya Math. J. 129 pp 43– (1993) · Zbl 0774.32016 · doi:10.1017/S0027763000004311 |
| [8] | DOI: 10.1007/PL00004754 · Zbl 0933.32048 · doi:10.1007/PL00004754 |
| [9] | DOI: 10.1090/S0002-9939-1962-0141795-0 · doi:10.1090/S0002-9939-1962-0141795-0 |
| [10] | DOI: 10.1007/s00209-004-0676-6 · Zbl 1066.32003 · doi:10.1007/s00209-004-0676-6 |
| [11] | (1991) |
| [12] | preprint |
| [13] | Dissertation (in Polish) (2004) |
| [14] | Trans. Amer. Math. Soc 354 (2002) |
| [15] | Univ. Iag. Acta Math. 38 pp 169– (2000) |
| [16] | Nagoya Math. J. 175 pp 165– (2004) · Zbl 1061.32010 · doi:10.1017/S0027763000008941 |
| [17] | Lectures on Functions of a Complex Variable pp 349– (1955) |
| [18] | Nagoya Math. J. 151 pp 221– (1998) · Zbl 0916.32016 · doi:10.1017/S0027763000025265 |
| [19] | DOI: 10.1090/S0002-9947-05-03738-4 · Zbl 1071.32008 · doi:10.1090/S0002-9947-05-03738-4 |
| [20] | DOI: 10.5802/afst.943 · Zbl 0963.32003 · doi:10.5802/afst.943 |
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