A Distortion Theorem for Quadrature Domains for Harmonic Functions

We prove that any finitely connected domain in the plane can be distorted so that it becomes ‘‘graviequivalent’’ to a signed measure with arbitrarily small support. Precisely: if D;C is a bounded, finitely connected domain with analytic boundary then for any agD and r)0, « )0 with B a, rq«.;D there exists a univalent function g in D with