Upper bound for distortion of capacity under conformal mapping

Abstract:

For a finitely-connected domain $\Omega$ containing $\infty$, with boundary $\Gamma$, the logarithmic capacity $d(\Gamma )$ is invariant under normalized conformal maps of $\Omega$. But the capacity of a subset $A \subset \Gamma$ will likely be distorted by such a map. Duren and Schiffer showed that the sharp lower bound for the distortion of the capacity of such a set is the so-called "Robin capacity" of the set $A$. We present here the sharp upper bound for the distortion, in terms of conformal invariants of $\Omega$: the harmonic measures of the boundary components of $\Omega$ and the periods of their harmonic conjugates (the Riemann matrix), and the capacity of $\Gamma$. In particular, the upper bound depends only on knowing which components of $\Gamma$ contain parts of $A$, not on the specific distribution of $A$. An extremal configuration is described explicitly for a special case.