Abstract

An important open problem in geometric complex analysis is to establish algorithms for explicit determination of the basic curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the new approaches and recent essential progress in this field of geometric complex analysis, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals.

Highlights

  • We are concerned with homeomorphisms reversing orientation

  • L ⊃ E with the same reflection coefficient; QE 1⁄4 min fQL : L ⊃ E quasicircleg: (4). The proof of this important theorem was given for finite sets E 1⁄4 fz1, ... , zng by Kühnau in [13], using Teichmüller’s theorem on extremal quasiconformal maps applied to the homotopy classes of homeomorphisms of the punctured spheres, and extended to arbitrary sets E ⊂ hC by the author in [14]

  • Ðz 1⁄4 x þ iyÞ: Let w0 ≔ wμ0 be an extremal representative of its class 1⁄2w0Š with dilatation kðw0Þ 1⁄4 ∥μ0∥∞ 1⁄4 inf fkðwμÞ : wμjL 1⁄4 w0jLg, and assume that there exists in this class a quasiconformal map w1 whose Beltrami coefficient μA1 satisfies the inequality ess supAr ∣μw1 ðzÞ∣ < kðw0Þ in some ring domain R 1⁄4 D ∗ nG complement to a domain G ⊃ D ∗

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Summary

Quasireflections and quasicurves

The classical Brouwer-Kerekjarto theorem ([1, 2], see [3]) says that every periodic homeomorphism of the sphere S2 is topologically equivalent to a rotation, or to a product of a rotation and a reflection across a diametral plane. The topological circles admitting such reflections are called quasicircles. Using a fractional linear transformation, one can send one of the points, for example, z4, to infinity; the above inequality assumes the form z2 À z1 ≤ C: z3 À z1 This is shown in [7] by applying the properties of quasisymmetric maps. Ahlfors has established that if a topological circle L admits quasireflections (i.e., is a quasicircle), there exists a differentiable quasireflection across L which is (euclidian) bilipschitz-continuous. This property is very useful in various applications. Quasireflections across more general sets E ⊂ ^ appear in certain questions and are of independent interest Those sets admitting quasireflections are called quasiconformal mirrors.

E E 1 À kE
Fredholm eigenvalues
The Grunsky and Milin inequalities
Extremal quasiconformality
Complex geometry and basic Finsler metrics on universal Teichmüller space
The Grunsky-Milin inequalities revised
The first Fredholm eigenvalue and Grunsky norm
Holomorphic motions
Main theorem
Examples
An open question
The main result
Some applications
Introductory remarks
There are unbounded convex polygons Pn for which the equalities (33) are valid in the strengthened form
Affine deformations and Grunsky norm
Scheme of the proof of Theorem 9
Generalization
Bounded polygons
An open problem here is the following question of Kühnau (personal communication)
General comments
Reflections across the finite collections of quasiintervals
Full Text
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