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 $\hat{\mathbb{C} }=\mathbb{C} \cup\{\infty\}$, the closed hyperbolic geodesic generating the fundamental group $π_1\big(\hat{\mathbb{C} }\setminus (E\cup F) \big) \cong \hat{\mathbb{Z} }$ is a $K$-quasicircle separating $E$ and $F$ with explicit distortion bound depending only on the capacity of $\hat{\mathbb{C} }\setminus (E\cup F)$. 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.