Grunsky inequalities and quasiconformal extension

Krushkal, Samuel L.; Kühnau, Reiner

doi:10.1007/bf02771975

Cited by 26 publications

(20 citation statements)

References 14 publications

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“…The following key results obtained in [Kr2], [Kr8] by applying the maps (2.6) underly the proofs of Theorems 1.1 and 1.2.…”

Section: Introductionmentioning

confidence: 90%

“…Now, applying to w ν ∈ Σ 0 (f µ 1 (D δ )) the variation of type (3.1), one obtains the following generalizations of (3.2) to Milin's coefficients given in [Kr8] …”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Its proof in [Kr2], [Kr8] shows that the functions (2.5) generate a maximizing sequence on which the Carathéodory distance c T (0, S f tµ 0 ) is attained and equals the Teichmüller distance (compare with Kra's theorem in [K] about the Carathéodory metric on Teichmüller abelian disks for Riemann surfaces with finitely generated fundamental groups).…”

Section: Introductionmentioning

confidence: 99%

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Milin's coefficients, complex geometry of Teichmüller spaces and variational calculus for univalent functions

Krushkal

2014

Georgian Mathematical Journal

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Abstract. We investigate the invariant metrics and complex geodesics in the universal Teichmüller space and Teichmüller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichmüller metric.Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory.2010 Mathematics Subject Classification: Primary: 30C55, 30C62, 30C75, 30F60, 32F45; Secondary: 30F45, 46G20Key words and phrases: Univalent, quasiconformal, Teichmüller space, infinite-dimensional holomorphy, invariant metrics, complex geodesic, Grunsky-Milin inequalities, variational problem, functional 1. Key theorems on invariant metrics and geodesics 1.1. Preamble. The Milin coefficient inequalities arose as a generalization of the classical Grunsky inequalities but coincide with the later only for conformal maps of the unit disk.We apply a quasiconformal variant of these inequalities to investigation of complex metric geometry and complex geodesics on two Teichmüller spaces: the universal space and Teichmüller space of the punctured disk and apply their geometry to variational calculus for univalent functions on the generic quasidisks with quasiconformal extensions. Such functions play an important role in the theory of Teichmüller spaces and also form one of the basic classes in geometric function theory.It will be shown that the intrinsic geometric features provide deep distortion results, in particular, allow one to solve explicitly some general variational problems. On the other hand, they cause surprising phenomena which do not arise in the classical variational theory for univalent functions.

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“…The following key results obtained in [Kr2], [Kr8] by applying the maps (2.6) underly the proofs of Theorems 1.1 and 1.2.…”

Section: Introductionmentioning

confidence: 90%

“…Now, applying to w ν ∈ Σ 0 (f µ 1 (D δ )) the variation of type (3.1), one obtains the following generalizations of (3.2) to Milin's coefficients given in [Kr8] …”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Milin's coefficients, complex geometry of Teichmüller spaces and variational calculus for univalent functions

Krushkal

2014

Georgian Mathematical Journal

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“…The minimum ( ) of dilatations ( ) = ‖ ‖ ∞ in the equivalence class of is called the Teichmüller norm of this function. It dominates the Grunsky norm [12] (or even a stronger form found recently in [9], but this will not be used here). These norms coincide only when any extremal Beltrami coefficient 0 for satisfies…”

Section: Introductionmentioning

confidence: 99%

A Question of Ahlfors

Krushkal

2014

Vestn. Volgogr. gos. univ., Ser. 1. Math. Phys.

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Abstract. In 1963, Ahlfors posed in [1] (and repeated in his book [2]) the following question which gave rise to various investigations of quasiconformal extendibility of univalent functions. Question. Let be a conformal map of the disk (or half-plane) onto a domain with quasiconformal boundary (quasicircle). How can this map be characterized?He conjectured that the characterization should be in analytic properties of the logarithmic derivative log ′ = ′′ / ′ , and indeed, many results on quasiconformal extensions of holomorphic maps have been established using ′′ / ′ and other invariants (see, e.g., the survey [9] and the references there).This question relates to another still not solved problem in geometric complex analysis: To what extent does the Riemann mapping function of a Jordan domain ⊂̂︀ C determine the geometric and conformal invariants (characteristics) of complementary domain* =̂︀ C ∖ ? The purpose of this paper is to provide a qualitative answer to these questions, which discovers how the inner features of biholomorphy determine the admissible bounds for quasiconformal dilatations and determine the Kobayashi distance for the corresponding points in the universal Teichmüller space.

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“…[12], [21], [19]). The functions with κ(f ) = k(f ) play a crucial role in applications of the Grunsky inequality technique and are characterized as follows.…”

Section: Grunsky Inequalities the Classical Grunsky Theorem States Tmentioning

confidence: 99%