On canonical metrics on Cartan-Hartogs domains

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain \(\Omega^{B^{d_0}}(\mu)\) endowed with the canonical metric \(g(\mu)\), we obtain an explicit formula for t...

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Veröffentlicht in:arXiv.org
Hauptverfasser: Feng, Zhiming, Tu, Zhenhan
Format: Paper
Sprache:Englisch
Veröffentlicht: Ithaca Cornell University Library, arXiv.org 31.03.2014
Schlagworte:
Analytic functions
Domains
Hilbert space
Thermal expansion
Weight
ISSN:2331-8422
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Zusammenfassung:The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain \(\Omega^{B^{d_0}}(\mu)\) endowed with the canonical metric \(g(\mu)\), we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal{H}_{\alpha}\) of square integrable holomorphic functions on \((\Omega^{B^{d_0}}(\mu), g(\mu))\) with the weight \(\exp\{-\alpha \varphi\}\) (where \(\varphi\) is a globally defined K\"{a}hler potential for \(g(\mu)\)) for \(\alpha>0\), and, furthermore, we give an explicit expression of the Rawnsley's \(\varepsilon\)-function expansion for \((\Omega^{B^{d_0}}(\mu), g(\mu)).\) Secondly, using the explicit expression of the Rawnsley's \(\varepsilon\)-function expansion, we show that the coefficient \(a_2\) of the Rawnsley's \(\varepsilon\)-function expansion for the Cartan-Hartogs domain \((\Omega^{B^{d_0}}(\mu), g(\mu))\) is constant on \(\Omega^{B^{d_0}}(\mu)\) if and only if \((\Omega^{B^{d_0}}(\mu), g(\mu))\) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.
Bibliographie:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1403.7975