Quasicircles as equipotential lines, homotopy classes and geodesics

Abstract

We give an application of our earlier results concerning the quasiconformal extension of a germ of a conformal map to establish that in two dimensions the equipotential level lines of a capacitor are quasicircles whose distortion depends only on the capacity and the level. As an application we find that given disjoint, nonseparating and nontrivial continua E and F in C = C ∪ {∞}, the closed hyperbolic geodesic generating the fundamental group π 1 Ĉ \ (E ∪ F)Z is a K-quasicircle separating E and F with explicit distortion bound depending only on the capacity ofĈ \ (E ∪ F). This result is then extended to obtain distortion bounds on a quasicircle representing a given homotopy class of a simple closed curve in a planar domain. Finally we are able to use these results to show that a simple closed hyperbolic geodesic in a planar domain is a quasicircle with a distortion bound depending explicitly, and only, on its length.