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Quasicircles as equipotential lines, homotopy classes and geodesics
Author(s) -
Martin Gaven J.
Publication year - 2016
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdw025
Subject(s) - mathematics , geodesic , distortion (music) , homotopy , simple (philosophy) , disjoint sets , hyperbolic geometry , pure mathematics , domain (mathematical analysis) , mathematical analysis , n connected , equipotential , equipotential surface , geometry , differential geometry , amplifier , philosophy , electrical engineering , cmos , epistemology , electronic engineering , engineering
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.