Condenser energy under holomorphic motions

Abstract

We prove that the condenser equilibrium energy is a superharmonic function when one of the plates of a condenser moves under a holomorphic motion and we characterize the cases where harmonicity occurs by showing that this happens if and only if the equilibrium measure on the moved plate is invariant under the holomorphic motion and the equilibrium measure on the fixed plate is unaffected. Also, in many cases, we show that harmonicity of the above function occurs only when the holomorphic motion is related with the level sets of the equilibrium potential of the condenser. In the case where the holomorphic motion is a dilation, we prove that harmonicity occurs if and only if the condenser is essentially an annulus with center at the origin.