The Distortion Theorem for quasiconformal mappings, Schottky’s Theorem and holomorphic motions

Martin, Gaven

doi:10.1090/s0002-9939-97-03677-0

Cited by 23 publications

(6 citation statements)

References 15 publications

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“…If more information is known -for instance if the capacity is known to be large -then better estimates are given in [6] and we will use these later, see (16). Now ψ −1 • η : D → Ω+ is quasiconformal with the bound at (7) and (7) and agrees with ϕ −1 on the boundary. In the same way we may construct f − : Ω − → Ĉ \ D with the same distortion bound and also agreeing with ϕ −1 on S. It is a moments work to now see that f : Ĉ → Ĉ defined by…”

Section: Proof Of Theoremmentioning

confidence: 52%

Quasicircles as equipotential lines, homotopy classes and geodesics

Martin

2016

Bull. London Math. Soc.

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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.

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Section: Proof Of Theoremmentioning

confidence: 52%

Quasicircles as equipotential lines, homotopy classes and geodesics

Martin

2016

Bull. London Math. Soc.

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“…See for example [31] and [18]. These two papers are related to a paper of Martin [19]. Furthermore, Martin worked in [20] on an extremal problem close to Teichmüller's one.…”

Section: Some Applicationsmentioning

confidence: 99%

A commentary on Teichmüller's paper "Verschiebungssatz der quasikonformen Abbildung" (A displacement theorem of quasiconformal mapping)

Alberge¹

2015

Preprint

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This is a commentary on Teichmüller's paper Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem of quasiconformal mapping), published in 1944. We explain in detail how Teichmüller solves the problem of finding the quasiconformal mapping from the unit disc to itself, sending 0 to a strictly negative point on the real line, holding the boundary of the disc pointwise fixed and with the smallest quasiconformal dilatation. We mention also some consequences of this extremal problem and we ask a question. PreliminariesAll along this paper, we shall be interested in planar quasiconformal mappings. Unless otherwise noted, all domains that we consider are connected subsets of the Riemann sphere C := C ∪ {∞}. There are several books which deal with quasiconformal mappings, see e.g. [5], [17] or [8].We give below two equivalent definitions of quasiconformal maps. Both of them are interesting and they introduce notions (module and quasiconformal dilatation) that Teichmüller used to solve Problem 1.1.A quadrilateral Q is a Jordan domain (i.e. a simply connected domain whose boundary is a Jordan curve) with four distinct boundary points. Sometimes, we will denote by Q (a, b, c, d) such a quadrilateral, where a, b, c and d are boundary points and we shall usually assume that these four points appear on the boundary in that order. By applying successively the Riemann Mapping Theorem, the Carathéodory Theorem 3 and a suitable Schwarz-Christoffel mapping, we know that Q is conformally equivalent (i.e. there exists a holomorphic bijection) to a rectangle 3 The theorem referred to is known as the boundary correspondance theorem.

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“…Then the following estimateR DA ≤ 1 16 e πKAholds, where K A is the quasiconformality coefficient of the mapping ϕ.Proof. Let r DA = minD(0,1) |ϕ −1 (x) − ϕ −1 (0)|.Then by the global distortion theorem[16] we haveR DA ≤116 e πKA r DA . Because |D(ϕ(0), r DA | ≤ |D A | = π we obtain an upper estimate for r DA ≤ 1.…”

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confidence: 99%

Principal frequencies of free non-homogeneous membranes and Sobolev extension operators

Gol’dshtein¹,

Pchelintsev²,

Ukhlov³

2021

Preprint

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Using the quasiconformal mappings theory and Sobolev extension operators, we obtain estimates of principal frequencies of free non-homogeneous membranes. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. Free nonhomogeneous membranes we consider as circular membranes in the quasiconformal geometry associated with non-homogeneity of membranes. As a consequence we get a connection between principal frequencies of free membranes and the smallest-circle problem (initially suggested by J. J. Sylvester in 1857).

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The Distortion Theorem for quasiconformal mappings, Schottky’s Theorem and holomorphic motions

Cited by 23 publications

References 15 publications

Quasicircles as equipotential lines, homotopy classes and geodesics

Quasicircles as equipotential lines, homotopy classes and geodesics

A commentary on Teichmüller's paper "Verschiebungssatz der quasikonformen Abbildung" (A displacement theorem of quasiconformal mapping)

Principal frequencies of free non-homogeneous membranes and Sobolev extension operators

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