The Bergman kernel and quadrature domains in the plane
arXiv:math/0403371
Abstract
A streamlined proof that the Bergman kernel associated to a quadrature domain in the plane must be algebraic will be given. A byproduct of the proof will be that the Bergman kernel is a rational function of z and one other explicit function known as the Schwarz function. Simplified proofs of several other well known facts about quadrature domains will fall out along the way. Finally, Bergman representative coordinates will be defined that make subtle alterations to a domain to convert it to a quadrature domain. In such coordinates, biholomorphic mappings become algebraic.
16 pages. To appear in a Birkhauser volume in honor of Harold Shapiro's 75th birthday. Research supported by NSF grant DMS-0305958