An explicit computation of the Bergman kernel function
type: Item
instance of: scholarly article
title: An explicit computation of the Bergman kernel function
zbMATH Open document ID: 0794.32021
DOI: 10.1007/BF02921591
author: John P. D'Angelo
published in: The Journal of Geometric Analysis
publication date: 1994-07-10T00:00:00Z
review text: We give an explicit computation of the Bergman kernel function on the domain \(\| z \|^ 2+\| w \|^{2p}<1\) contained in complex Euclidean space of dimension \(n+m\). Here \(p\) is any positive number. We obtain the exact formula \[ K \bigl( (z,w), \overline {(z,w)} \bigr)=\sum^{n+1}_{k=0} c_ k {\bigl( 1-\| z \|^ 2)^{-n- 1+{k \over p}} \over \biggl( \bigl( 1-\| z \|^ 2 \bigr)^{{1 \over p}}-\| w \|^ 2 \biggr)^{m+k}}. \] The constants \(c_ k\) depend on \(k,n,m,p\) and we compute them. Writing \(d_{k+1} = \pi^{n+m} p^ n {c_{k+1} \over \Gamma (m+k+1)}\) we obtain the formula \[ \prod^ n_{l=0} (y+pl)=\sum^ n_{k=-1} d_{k+1} \prod^ k_{l=0} (y+l) \] and determine \(d_ j\) by evaluating the parameter \(y\) at the negative integers. For example \[ d_ 1=-\prod^ n_{l=0} (pl-1)=\prod^ n_{l=1} (pl-1) \quad \text{and } d_ 2={1 \over 2} \prod^ n_{l=0} (pl-2)-\prod^ n_{l=0} (pl-1). \]
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 579773
zbMATH Keywords: weakly pseudoconvex
zbMATH Keywords: beta function
zbMATH Keywords: Bergman kernel function
MaRDI profile type: Publication
cites work: A note on the Bergman kernel
cites work: Several Complex Variables and the Geometry of Real Hypersurfaces
cites work: Local analyticity for the \(\bar {\partial{}}\)-Neumann problem and \({\square{}}_ b\) -- some model domains without maximal estimates
cites work: Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results
cites work: The Bergman kernel and biholomorphic mappings of pseudoconvex domains
cites work: Deformation of complex structures, estimates for the (partial d) equation, and stability of the Bergman kernel
cites work: The growth of the bergman kernel on pseudoconvex domains of homogeneous finite diagonal type
cites work: Asymptotic behavior of the curvature of the Bergman metric of the thin domains
cites work: Lower bounds on the Bergman metric near a point of finite type
full work available at URL: https://doi.org/10.1007/bf02921591
OpenAlex ID: W2136505896
description: scientific article; zbMATH DE number 579773