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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

New formulas of the Bergman kernels for complex ellipsoids in $ \mathbb{C}^2$


Author: Jong-Do Park
Journal: Proc. Amer. Math. Soc. 136 (2008), 4211-4221
MSC (2000): Primary 32A25; Secondary 33D70
Published electronically: July 15, 2008
MathSciNet review: 2431034
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Abstract: We compute the explicit formula of the Bergman kernel for a nonhomogeneous domain $ \{(z_1,z_2)\in\mathbb{C}^2:\vert z_1\vert^{4/q_1}+\vert z_2\vert^{4/q_2}<1\}$ for any positive integers $ q_1$ and $ q_2$. We also prove that among the domains $ D_p:=\{(z_1,z_2)\in\mathbb{C}^2:\vert z_1\vert^{2p_1}+\vert z_2\vert^{2p_2}<1\}$ in $ \mathbb{C}^2$ with $ p=(p_1,p_2)\in\mathbb{N}^2$, the Bergman kernel is represented in terms of closed forms if and only if $ p=(p_1,1),(1,p_2)$, or $ p=(2,2)$.


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Additional Information

Jong-Do Park
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Address at time of publication: Department of Mathematics, Pohang University of Science and Technology, San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk, 790-784, Korea
Email: jongdopark@gmail.com, jdpark@postech.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09576-2
PII: S 0002-9939(08)09576-2
Keywords: Bergman kernel, homogeneous domains, hypergeometric function, complex ellipsoids
Received by editor(s): February 28, 2007
Published electronically: July 15, 2008
Additional Notes: The author was supported by Korea Research Foundation Grant 2005-070-C00007 and partially supported by BK21 CoDiMaRO
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.