Abstract
The Fock–Bargmann–Hartogs domain \(D_{n,m}(\mu )\) (\(\mu >0\)) in \(\mathbb {C}^{n+m}\) is defined by the inequality \(\Vert w\Vert ^2<e^{-\mu \Vert z\Vert ^2},\) where \((z,w)\in \mathbb {C}^n\times \mathbb {C}^m\), which is an unbounded non-hyperbolic domain in \(\mathbb {C}^{n+m}\). This paper introduces a Kähler metric \(\alpha g(\mu ;\nu )\) \((\alpha >0)\) on \(D_{n,m}(\mu )\), where \(g(\mu ;\nu )\) is the Kähler metric associated with the Kähler potential \(\Phi (z,w):=\mu \nu {\Vert z\Vert }^{2}-\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)\) (\(\nu >-1\)) on \(D_{n,m}(\mu )\). The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on \((D_{n,m}(\mu ), g(\mu ;\nu ))\) with the weight \(\exp \{-\alpha \Phi \}\) for \(\alpha >0\). Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric \(\alpha g(\mu ;\nu )\) \((\alpha >0)\) on the domain \(D_{n,m}(\mu )\) to be a balanced metric. So, we obtain the existence of balanced metrics for a class of Fock–Bargmann–Hartogs domains.
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Acknowledgments
E. Bi was supported by the Fundamental Reasearch Funds for the Central Universities (No. 2014201020204), Z. Feng was supported by the Scientific Research Fund of Leshan Normal University (No. Z1513), and Z. Tu was supported by the National Natural Science Foundation of China (No. 11271291).
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Bi, E., Feng, Z. & Tu, Z. Balanced metrics on the Fock–Bargmann–Hartogs domains. Ann Glob Anal Geom 49, 349–359 (2016). https://doi.org/10.1007/s10455-016-9495-3
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DOI: https://doi.org/10.1007/s10455-016-9495-3