Characterization of quasi-disks and Teichmüller spaces
Suppose \(\Gamma\) is a finitely generated Fuchsian group of the first kind acting on the upper half-plane U. Let T(\(\Gamma)\) denote the Bers embedding of the Teichmüller space of \(\Gamma\) into the Banach space \(B_ 2(L,\Gamma)\) of holomorphic quadratic differentials on the lower half-plane L which are bounded when divided by the square of the hyperbolic metric on L. Let S(\(\Gamma)\) be the set of all Schwarzian derivatives of univalent meromorphic functions on L. It is known that S(\(\Gamma)\) is closed and \(T(\Gamma)\cup \partial T(\Gamma)\subset S(\Gamma).\) The author proves that the interior of S(\(\Gamma)\) in \(B_ 2(L,\Gamma)\) is connected and equal to T(\(\Gamma)\). Also, he shows that the boundary of T(\(\Gamma)\) in \(B_ 2(L,\Gamma)\) can be very complicated by presenting a negative answer to a conjecture of Bers. In addition, the author discusses quasi-disks. P. Jones proved that a simply connected region \(\Omega\) in the complex plane \({\mathbb{C}}\) is a quasi-disk if and only if \(\Omega\) has the BMO extension property (that is, there is a constant \(M>0\) such that for all \(f\in BMO(\Omega)\) there exists \(F\in BMO({\mathbb{C}})\) such that \(\| F\|_{BMO({\mathbb{C}})}\leq M \| f\|_{BMO(\Omega)}).\) The author proves that it is sufficient to verify the BMO extension property just for the subclass ABD(\(\Omega)\) of bounded holomorphic functions on \(\Omega\) having finite Dirichlet integral. He also establishes an analog for finitely connected Jordan regions.
scientific article; zbMATH DE number 3933478
scientific article