Critical measures and extremal compact sets

We give the definition of measures on the complex plane that are critical for energy functionals with respect to a wide class of kernels including logarithmic and Green kernels with external fields. We show the connection of equlibrium critical measures with meromorphic quadratic differentials in domains that are defined by the kernels. We use this connection for description of the structure of compact sets of minimal capacity that arise in some problems of complex analysis.