Quadrature domains in $\mathbb C^n$

TL;DR

This paper establishes that quadrature domains are densely approximable within certain classes of product and Hartogs domains in complex spaces, advancing understanding of their geometric and functional properties.

Contribution

It proves two density theorems showing quadrature domains are dense in specific classes of product and Hartogs domains in complex spaces.

Findings

Quadrature domains are dense in product domains of the form D×Ω with certain smoothness conditions.

Quadrature domains are dense in smoothly bounded complete Hartogs domains in C^2.

The results extend the understanding of the geometric distribution of quadrature domains in complex analysis.

Abstract

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 $D \times \Omega$, 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 $\mathbb{C}^2$.