Elliptic, hyperbolic, and condenser capacity; geometric estimates for elliptic capacity

The elliptic capacity of a compact setE in the plane is defined by R. Kühnau by least energy considerations analogous to the definition of logarithmic capacity but using the elliptic distance in place of the euclidean distance. We prove an identity connecting elliptic capacity, extremal length, and the capacity of the condenser with platesE andE *, whereE * is antipodal toE in the Riemann sphere. An analogous identity for hyperbolic capacity, conjectured by P. Duren and J. Pfaltzgraff, is also established. We show that elliptic capacity is always smaller than hyperbolic capacity, thus answering a question of P. Duren and R. Kühnau. We prove some geometric estimates for elliptic capacity. These estimates involve geometric transformations such as polarization, circular symmetrization, projection, and shoving.