Measures that define a compact Cauchy transform

The aim of this work is to provide a geometric characterization of the positive Radon measures $\mu$ with compact support on the plane such that the associated Cauchy transform defines a compact operator from $L^2(\mu )$ to $L^2(\mu )\mathrm{.}$ It turns out that a crucial role is played by the density of the measure and by its Menger curvature.