Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Katsuhiko Matsuzaki

doi:10.4171/RMI/1169

Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Katsuhiko Matsuzaki^*

^*Corresponding author for this work

School of Education

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with α-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C^1+α-topology at the base point.

Original language	English
Pages (from-to)	1333-1374
Number of pages	42
Journal	Revista Matematica Iberoamericana
Volume	36
Issue number	5
DOIs	https://doi.org/10.4171/RMI/1169
Publication status	Published - 2020 Feb 10

Keywords

Beltrami coefficients
Bers embedding
Circle diffeomorphism
Hölder continuous derivative
Quasiconformal map
Quasisymmetric homeomorphism
Schwarzian derivative
Universal Teichmüller space

ASJC Scopus subject areas

Mathematics(all)

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Cite this

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title = "Teichm{\"u}ller space of circle diffeomorphisms with H{\"o}lder continuous derivatives",

abstract = "Based on the quasiconformal theory of the universal Teichm{\"u}ller space, we introduce the Teichm{\"u}ller space of diffeomorphisms of the unit circle with α-H{\"o}lder continuous derivatives as a subspace of the universal Teichm{\"u}ller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C1+α-topology at the base point.",

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author = "Katsuhiko Matsuzaki",

note = "Funding Information: This work was supported by JSPS KAKENHI 25287021. Publisher Copyright: {\textcopyright} European Mathematical Society.",

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doi = "10.4171/RMI/1169",

language = "English",

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T1 - Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

AU - Matsuzaki, Katsuhiko

N1 - Funding Information: This work was supported by JSPS KAKENHI 25287021. Publisher Copyright: © European Mathematical Society.

PY - 2020/2/10

Y1 - 2020/2/10

N2 - Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with α-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C1+α-topology at the base point.

AB - Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with α-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C1+α-topology at the base point.

KW - Beltrami coefficients

KW - Bers embedding

KW - Circle diffeomorphism

KW - Hölder continuous derivative

KW - Quasiconformal map

KW - Quasisymmetric homeomorphism

KW - Schwarzian derivative

KW - Universal Teichmüller space

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DO - 10.4171/RMI/1169

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JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

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ER -

Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Abstract

Keywords

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