Quadrature domains and kernel function zipping

Abstract

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be zipped down to a strikingly small data set. It is also proved that the kernel functions associated to a quadrature domain must be algebraic.Comment: 13 pages, to appear in Arkiv for matemati

Random matrices in 2D, Laplacian growth and operator theory

“…This is clearly demonstrated in the following theorem due to M. Sakai [137], [138]. The second part of the theorem is discussed (and proved) in some other forms also in [108], [139], [106], [107], [140], for example:…”

Selected topics on quadrature domains

“…This is clearly demonstrated in the following theorem due to M. Sakai [70], [71]. The second part of the theorem is discussed (and proved) in some other forms also in [16], [20], [4], [5], [85], for example.…”

Harmonic balls and the two-phase Schwarz function

“…1) Moment problems, Operator Theory (hyponormal operators), Exponential Transforms (see [12,13,20,21,22,23,24,37,38,39], 2) Hele-Shaw related problems (see [5,17,18]), 3) Problems related to Bergman and Szegö Kernels (see [3,4]), 4) Mother bodies and skeletons (see [11,15,28,36], 5) The Cauchy Problem in C n (see [7,19,34]), 6) Quadrature Surfaces (see [30,31] . It is tantalizing and wishful to think that many of these concepts can be developed also in the two-phase situation.…”