Quasicircles as equipotential lines, homotopy classes and geodesics

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 inℂ ^ = ℂ ∪ { ∞ } , the closed hyperbolic geodesic generating the fundamental groupπ 1 ( ℂ ^ ∖ ( E ∪ F ) ) ≅ ℤ is a K ‐quasicircle separating E and F with explicit distortion bound depending only on the capacity ofC ^A A∖ ( 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.