Quadrature domains in $\mathbb C^n$

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

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