Explicit formulas of the Bergman kernel for 3-dimensional complex ellipsoids
type: Item
instance of: scholarly article
title: Explicit formulas of the Bergman kernel for 3-dimensional complex ellipsoids
zbMATH Open document ID: 1267.32001
author: Jong-Do Park
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-03-11T00:00:00Z
review text: The author considers the complex ellipsoids \[ \Omega _{p_1, \dots , p_n} = \bigg\{ (z_1, \dots , z_n) \in \mathbb C^n : \sum _{j=1}^n |z_j|^{2p_j} < 1 \bigg\} \] where each \(p_j\) is a positive real number. The aim is to obtain an explicit formula for the Bergman kernel for such ellipsoids in closed form, i.e., without writing down an infinite series. Because of difficulties, results of this type were obtained up to now by several authors only for a rather limited set of domains \(\Omega _{p_1, \dots , p_n}\). The main theorems of the present paper are: Theorem 1.1. The Bergman kernel for \(\big\{ (z_1, z_2, z_3) \in \mathbb C^3: |z_1|^4+ |z_2|^4+ |z_3|^4 <1 \big\}\) is given by \[ \begin{multlined} K(z,w) = \frac {48 \eta _1 \eta _2 \eta _3}{\pi ^3(1- \eta _1^2 - \eta _2^2 - \eta _3^2)^4} + \frac {48 (1+\eta _1 +\eta _2 +\eta _3 )}{\pi ^4(1- \eta _1^2 - \eta _2^2 - \eta _3^2)^3}\\ +\frac 6{\pi ^4}\diamondsuit _3 \bigg\{ \sum _{(i,j,k)\in J} \frac {c'_{ijk}(1+\eta _1 + \eta _2)^{1-k}(1+\eta _1)^k} {(1-\eta _1^2 -\eta _2^2 -\eta _3^2 )^i(1-\eta _1^2 -\eta _2^2 )^j (1-\eta _1^2)^k} \bigg\} \\ + \frac {3}{\pi ^4 (1-\eta _1^2 - \eta _2^2 - \eta _3^2 )^4 } \diamondsuit _3 \bigg( \frac {\eta _1 \eta _2 \sum _{i+j=3}c_{ij}(1-\eta _1 ^2 - \eta _2 ^2 )^i (\eta _3^2)^j} {(1-\eta _1^2-\eta _2^2)^{\frac 52}} \\ \times \bigg( \pi + 2\arcsin \frac {\eta _1}{\sqrt{1-\eta _2^2}} +2 \arcsin \frac {\eta _2} {\sqrt{1-\eta _1^2}} +2 \arcsin \frac {\eta _1 \eta _2}{\sqrt{1-\eta _1^2}\sqrt{1-\eta _2^2}}\bigg)\bigg) \end{multlined} \] where \(\eta _j = z_j\overline w_j\) for \(j=1,2,3\), \(\diamondsuit _3\) is defined by \[ \diamondsuit _3 f(\eta _1, \eta _2, \eta _3) = f(\eta _1, \eta _2, \eta _3) + f(\eta _2, \eta _3, \eta _1) + f(\eta _3, \eta _1, \eta _2), \] \(J = \{ (2,1,0), (1,2,0), (1,1,1) \} \) and \(c'_{210}= -2\), \(c'_{111} = \frac 13 \), \(c_{30}=5\), \(c_{21}=15\), \(c_{12}=-5\), \(c_{03}=1\). Then the author states Theorem 1.2, in which a closed form of the Bergman kernel…
reviewed by: Piotr Jakobczak
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32T25
Mathematics Subject Classification ID: 33C65
zbMATH DE Number: 6143243
zbMATH Keywords: complex ellipsoids
zbMATH Keywords: Bergman kernel
zbMATH Keywords: hypergeometric functions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.11.017
OpenAlex ID: W2075440253
DOI: 10.1016/J.JMAA.2012.11.017
description: scientific article; zbMATH DE number 6143243