The Green's Function, the Bergman Kernel and Quadrature Domains in Cn

Abstract

In the first part of this thesis, we prove two density theorems for quadrature domains in Cn ,n≥2. It is shown that quadrature domains are dense in the class of all product domains of the form D×Ωwhere D⊂Cn−1 is a smoothly bounded pseudoconvex domain satisfying Bell’s Condition R and Ω⊂Cis a smoothly bounded domain. It is also shown that quadrature domains are dense in the class of all smoothly bounded complete Hartogs domains in C2. In the second part of this thesis, we study the behaviour of the critical points of the Green’s function when a sequence of domains Dk⊂Rn con-verges to a limiting domain Din the C∞-topology. It is shown that the limit-ing set of the critical points of the Green’s functions Gkfor domains Dk⊂Care the zeroes of the Bergman kernel of D. This generalizes a result of Solynin and Gustafsson, Sebbar.