Capacity Functions

Capacity functions were born out of geometric. necessity, a decade and a half ago. Plane regions had been found of arbitrarily small area, yet with a totally disconnected boundary. Such regions seemed to defy the very spirit of Riemann's mapping theorem. They could be mapped conformally and univalently into a disk, with the single boundary point at infinity being stretched into a circle. The plausible explanation of the mystery is, of course, as follows. Under a mapping of the punctured sphere onto a disk, an area element near the punctured point would have to stretch more in the circular direction than in the radial direction, and the conformality would be destroyed. But if there is a sufficiently heavy accumulation of other boundary components, these can take over the distortion, and the mapping of the region itself remains conformal. Such phenomena made it an important problem to characterize pointlike boundary components which were unstable, i.e., hid in them­ selves this power of stretching into proper continua. Standard tools such as mass distributions, potentials, and transfinite diameters could not be used here, as they were subject to the vagaries of the other com­ ponents. The characterization had to be intrinsic, depending only on the region itself, in a conformally invariant manner. This goal was achieved in the following fashion (SARlO [10, 13]).