Title A remark on quadratic differentials vanishing at infinity(Complex ...

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A remark on quadratic differentials vanishing at infinity(Complex Analysis and Geometry of Hyperbolic Spaces) MATSUZAKI, KATSUHIKO

数理解析研究所講究録 (2006), 1518: 144-145

2006-10

http://hdl.handle.net/2433/58729

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Departmental Bulletin Paper

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Kyoto University

数理解析研究所講究録 1518 巻 2006 年 144-145

144

A remark on quadratic differentials vanishing at infinity

KATSUHIKO

$\mathrm{L}\mathrm{I}44\mathrm{T}\mathrm{S}1j^{r}\angle \mathrm{A}\mathrm{K}\mathrm{I}$

松崎克彦

Okayama University 岡山人四白然科学研究科 For a Fuchsian group acting on tlxc upper half-plane H. let denote the Banach space of all holomorphic functions on $H$ satisfying for every , where and ancl . An clcincnt in is called a bounded holomorphic quadratic differential for . Let be a subset of consisting of thosc . whcre is the Schwarzian derivative a F-compatiblc univalent function on $H$ . The Nebari theorem says that if thcn . Also is closed in . boundary s’elni-norln for is defined by $\Gamma$

$B(\Gamma)$

$\gamma^{*}\varphi=(\rho$

$||\varphi||0$ and all $r>1$ . Assume that for $con,formal$ automorphisms $H$ of such that th. orbit for any uompact subset $V\subset H$ and such that Then there cxists with

$Lct,\tilde{\psi}\in B(\Gamma)satisfi_{J}||\tilde{\phi}||.(3(1’)0$

$\varphi\in B(\Gamma)$

$r\phi\not\in S(\Gamma)$

$||\varphi||\underline{((\grave{)})}(1’)0$

$\mathrm{t}^{\backslash },\mathrm{t}\mathrm{l}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}’\subset H.$

$h_{r\iota_{k}}$

$\tilde{\phi}=.\sum_{k=1}^{\infty}(h^{-1})^{*}|\iota\iota\cdot\phi\in B(\Gamma)$

satisfies Thcn we can applv the

. and so that

$(||\tilde{\psi}||\backslash \backslash ^{\mathrm{t})1(1’)}\leq)||\tilde{\phi}||