Quasicircles as equipotential lines, homotopy classes and geodesics

TL;DR

This paper demonstrates that equipotential lines of capacitors and certain hyperbolic geodesics in planar domains are quasicircles with distortion bounds depending solely on capacity or length, extending quasiconformal mapping theory.

Contribution

It establishes explicit distortion bounds for quasicircles arising from equipotential lines, hyperbolic geodesics, and homotopy classes in planar domains, based on capacity and length.

Findings

Equipotential lines are quasicircles with distortion depending on capacity and level.

Hyperbolic geodesics generating fundamental groups are quasicircles with explicit distortion bounds.

Distortion bounds for quasicircles representing homotopy classes depend only on capacity or length.

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 $\pi_1\big(\hat{\mathbb{C} }\setminus (E\cup F) \big) \cong \hat{\mathbb{Z} }$ 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.