Multivalued analytic continuation of the Cauchy transform

In this paper we introduce the notion of multivalued analytic continuation of the Cauchy transforms. Many difficulties arise because the continuation is not single-valued. Our main result asserts that if chi Omega has a multivalued analytic continuation, then the free boundary partial derivative Omega has zero Lebesgue measure. Here chi Omega is the characteristic function of a domain Omega and partial derivative Omega is its boundary. We also discuss the connections between this notion, quadrature domains and approximations of analytic functions with single-valued integrals by rational functions. The last problem is related to the existence of a continuous function g and a closed connected set K such that the gradient of g vanishes on K, nevertheless g is not constant on K.