We prove two density theorems for quadrature domains in C n , n ≥ 2. It is shown that quadrature domains are dense in the class of all product domains of the form D × Ω, where D ⊂ C n−1 is a smoothly bounded domain satisfying Bell's Condition R and Ω ⊂ C is a smoothly bounded domain and also in the class of all smoothly bounded complete Hartogs domains in C 2 .

Abstract. We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and L 2 -cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green's function and related objects.

The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of a quasi-balanced domain is in fact smooth away from the origin. This allows us to construct a smooth plurisubharmonic defining function for such domains. Our result is new even in the case of balanced domains.2010 Mathematics Subject Classification. Primary 32A07.

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It is known that if f : D1 → D2 is a polynomial biholomorphism with polynomial inverse and constant Jacobian then D1 is a 1-point Quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever D2 is a balanced domain. Bell conjectured that all 1-point Quadrature domains arise in this manner. In this note, we construct a 1-point Quadrature domain of order 1 that is not biholomorphic to any balanced domain. arXiv:1807.08689v1 [math.CV]