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On the first two coefficients of the Bergman function expansion for radial metrics. (English) Zbl 1369.32016

Summary: Let \(g_F\) be Kähler metrics on rotation invariant domains \(\varOmega = \mathbb{B}^d, \mathbb{C}^d, \mathbb{B}^{d \ast}, \mathbb{C}^{d \ast}\) associated with the Kähler potentials \(\varPhi_F(z, \overline{z}) = F(\ln(\| z \|^2))\). The purpose of this paper is twofold. Firstly, we obtain explicit formulas of the coefficients \(a_j\) (\(j = 1, 2\)) of the Bergman function expansion for the domains \((\varOmega, g_F)\). Secondly, we obtain explicit expressions of \(F\) when both \(a_1\) and \(a_2\) are constants on \(\varOmega\).

MSC:

32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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