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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Jordan domains and the universal Teichmüller space

Author: Barbara Brown Flinn
Journal: Trans. Amer. Math. Soc. 282 (1984), 603-610
MSC: Primary 30C60; Secondary 30C35, 32G15
MathSciNet review: 732109
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Abstract: Let $ L$ denote the lower half plane and let $ B(L)$ denote the Banach space of analytic functions $ f$ in $ L$ with $ {\left\Vert f \right\Vert _L} < \infty $, where $ {\left\Vert f \right\Vert _L}$ is the suprenum over $ z \in L$ of the values $ \left\vert {f(z)} \right\vert{(text{Im} z)^2}$. The universal Teichmüller space, $ T$, is the subset of $ B(L)$ consisting of the Schwarzian derivatives of conformal mappings of $ L$ which have quasiconformal extensions to the extended plane. We denote by $ J$ the set

$\displaystyle \left\{ {{S_f}:f{\text{is conformal in }}L{\text{and }}f(L){\text{is a Jordan domain}}} \right\},$

which is a subset of $ B(L)$ contained in the Schwarzian space $ S$. In showing $ S - \bar T \ne \emptyset $, Gehring actually proves $ S - \bar J \ne \emptyset $. We give an example which demonstrates that $ J - \bar T \ne \emptyset $.

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Additional Information

PII: S 0002-9947(1984)0732109-2
Keywords: Schwarzian derivative, quasicircle, universal Teichmüller space
Article copyright: © Copyright 1984 American Mathematical Society