The Grunsky Coefficients as a Model of Universal Teichmüller Space

Krushkal, Samuel L.

doi:10.22606/aan.2017.23006

Cited by 3 publications

(2 citation statements)

References 10 publications

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“…We mention here another model constructed in [25] by applying the Grunsky coefficients of univalent functions in the disk. In this model, the space T is represented by a bounded domain in a subspace of l ∞ determined by the Grunsky coefficients.…”

Section: Modeling Universal Teichm üLler Space By Grunski Coefficientsmentioning

confidence: 99%

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Grunsky operator, Grinshpan's conjecture and universal Teichmuller space

Krushkal¹

2022

Preprint

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A. Grinshpan posed a deep conjecture on the norm of the Grunsky operator generated by univalent functions in the disk. It gives a quantitative answer in terms of the Grunsky coefficients, to which extent a univalent function determines the bound of dilatations of its quasiconformal extensions. We provide the proof of this conjecture and its various analytic, geometric and potential applications.Another result concerns the model of universal Teichmüller space by Grunsky coefficients.

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Section: Modeling Universal Teichm üLler Space By Grunski Coefficientsmentioning

confidence: 99%

“…It is established in [25] that the sequences c corresponding to functions f ∈ Σ 0 having quasiconformal extensions to the disk D fill a bounded domain T in the indicated Banach space L containing the origin, and the correspondence…”

Section: Modeling Universal Teichm üLler Space By Grunski Coefficientsmentioning

confidence: 99%

Grunsky operator, Grinshpan's conjecture and universal Teichmuller space

Krushkal¹

2022

Preprint

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Fredholm Eigenvalues and Quasiconformal Geometry of Polygons

Krushkal

2021

J Math Sci

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Fredholm eigenvalues and quasiconformal geometry of polygons

Krushkal

2020

UMB

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An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but 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 newapproaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

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The Grunsky Coefficients as a Model of Universal Teichmüller Space

Cited by 3 publications

References 10 publications

Grunsky operator, Grinshpan's conjecture and universal Teichmuller space

Grunsky operator, Grinshpan's conjecture and universal Teichmuller space

Fredholm Eigenvalues and Quasiconformal Geometry of Polygons

Fredholm eigenvalues and quasiconformal geometry of polygons

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