Complex Geometry of the Universal Teichmuller Space

Krushkal, Samuel L.

doi:10.1023/b:simj.0000035830.46662.75

Cited by 7 publications

(2 citation statements)

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“…The holomorphy of the Grunsky map was stated without the proof by Krushkal (see [2, p. 303]) and was frequently used in his later works (see [3][4][5][6][7][8][9][10][11]), while the holomorphy of the Grunsky coefficients was stated without the proof by Shiga-Tanigawa (see [14, p. 363]). Since the Grunsky operator plays an important role in the study of the Teichmüller spaces (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]), here we give complete proofs of these results into literature. A map similar to the Grunsky map was introduced and proved to be holomorphic by Takhtajan-Teo (see [31]).…”

Section: Holomorphy Of Grunsky Mapmentioning

confidence: 99%

“…The above results can be restated in the terminologies from the functional analysis. First note that l 2 is a Hilbert space of sequences x = (x m ) with the inner product and the norm It is well known that the Grunsky operator plays an important role in the univalent function theory (see [1]) and in the study of the Teichmüller spaces (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]). In this note we will discuss the dependence of the Grunsky operator on a univalent function.…”

Section: Introductionmentioning

confidence: 99%

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On Grunsky operator

Shen

2007

Sci. China Ser. A-Math.

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Section: Holomorphy Of Grunsky Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Grunsky operator

Shen

2007

Sci. China Ser. A-Math.

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Grunsky inequalities and quasiconformal extension

Krushkal

Kühnau

2006

Isr. J. Math.

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Hyperbolic metrics on universal Teichmüller space and extremal problems

Krushkal

2012

J Math Sci

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The paper consists of two parts, both related to the complex geometry of the universal Teichmüller space. We reprove that all contractible invariant metrics on this space coincide and apply this important fact to solving the general extremal problems for univalent functions with quasiconformal extensions.The complex metric geometry of the universal Teichmüller space causes some interesting phenomena in the classical variational problems for univalent functions. In this paper, we investigate an interaction between these topics.First, we reprove that all contractible invariant metrics on the universal Teichmüller space agree with its intrinsic Teichmüller metric.Theorem 1.1. The Carathéodory metric of the universal Teichmüller space T coincides with its Kobayashi metric, hence all invariant metrics on T are equal to the Teichmüller metric.The infinitesimal forms of these metrics coincide with the canonical Finsler structure on T that generates the Teichmüller metric.This important fact, established in [14] by the renormalization of sequences of holomorphic quadratic differentials generating the extremal Beltrami coefficients, underlies various profound applications. Its new proof is much simpler and relies on some results that have intrinsic interest. Note that such a result is still unknown for other Teichmüller spaces.Theorem 1.1 is deeply connected with general extremal problems and the distortion theorem for univalent functions with quasiconformal extensions and gives rise to their strengthening. We consider these problems in the last section. BackgroundHere, we briefly present certain results underlying the proof of Theorem 1.1. This exposition is adapted to our special cases. Let ∆ = {|z| < 1}, ∆ * = C \ ∆.

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Complex Geometry of the Universal Teichmuller Space

Cited by 7 publications

References 40 publications

On Grunsky operator

On Grunsky operator

Grunsky inequalities and quasiconformal extension

Hyperbolic metrics on universal Teichmüller space and extremal problems

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