Uniqueness of simply connected quadrature domains from their quadrature function

Establish that every simply connected quadrature domain in the complex plane is uniquely determined by its rational quadrature function up to the conformal radius; specifically, show that if two simply connected domains Ω1 and Ω2 satisfy Ωi ∈ QD(h) for the same rational quadrature function h, then Ω1 and Ω2 coincide after the normalization fixing the conformal radius (i.e., they are identical modulo scaling by the conformal radius).

Background

The paper reviews that quadrature domains (QDs) are characterized by admitting exact quadrature identities and having an associated rational quadrature function h. While uniqueness fails in general for multiply connected regions (e.g., distinct domains sharing the same h), the authors highlight a conjectural uniqueness in the simply connected case. They note classical results establishing uniqueness for particular bounded finitely connected cases (such as disks and cardioids), and point to recent literature discussing further uniqueness results.

The conjecture seeks to resolve the inverse problem of determining whether the quadrature function h identifies a unique simply connected QD up to the natural normalization by conformal radius, thereby settling a central question about the structure of the correspondence between h and Ω in the simply connected setting.