Quadrature domains in ${\mathbb{C}}^n$

Abstract

We prove two density theorems for quadrature domains in ${\mathbb{C}}^n$, $n\ge 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D\times \Omega$, where $D\subset {\mathbb{C}}^{n-1}$ is a smoothly bounded domain satisfying Bell's Condition R and $\Omega \subset \mathbb{C}$ is a smoothly bounded domain and also in the class of all smoothly bounded complete Hartogs domains in ${\mathbb{C}}^2$.

13 pages