Abstract: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 χ has a multivalued analytic continuation, then the free boundary ∂ has zero Lebesgue measure. Here χ is the characteristic function of a domain and ∂ is its boundary. We also discuss the connections between this notion, quadrature domains and approximations of analytic functions with single-valued integrals b… Show more
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