On canonical metrics on Cartan–Hartogs domains
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- 作者：Zhiming Feng ; Zhenhan Tu
- 关键词：Bounded symmetric domains ; Cartan–Hartogs domains ; Bergman kernels ; K?hler metrics ; 32A25 ; 32M15 ; 32Q15
- 刊名：Mathematische Zeitschrift
- 出版年：2014
- 出版时间：October 2014
- 年：2014
- 卷：278
- 期：1-2
- 页码：301-320
- 全文大小：295 KB
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Zhenhan Tu (2)
1. School of Mathematical and Information Sciences, Leshan Normal University, Leshan, 614000, Sichuan, People’s Republic of China
2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, Hubei, People’s Republic of China - ISSN：1432-1823
文摘
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 \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined K?hler potential for \(g(\mu )\) ) for \(\alpha >0\) , and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) 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 \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.