On the space of schlicht projective structures on compact Riemann surfaces with boundary
Let \(\Gamma\) be a Fuchsian group; to \(\Gamma\) we associate the totality of univalent holomorphic functions \(f\) for which \(f\circ\gamma=\chi(\gamma)\circ f\) for \(\gamma\in\Gamma\) and where \(\chi(\gamma)\) is a Möbius transformation. We denote by \(S(\Gamma)\) the Schwarzian derivatives of all of these functions; \(S(\Gamma)\) can be considered as a subset of the space of quadratic differentials of bounded norm. Likewise one can represent the Teichmüller space \(T(\Gamma)\) of \(\Gamma\) as a subset of the same space, and it is, by the Bers' construction, a subset of \(S(\Gamma)\). The purpose of this paper is to show that \(T(\Gamma)=\text{Int}(S(\Gamma))\) if \(\Gamma\) is purely hyperbolic of the second kind. The analogous result for groups of the first kind is due to H. Shiga. It is worth noting in this connection that, in general, \(S(\Gamma)\) is larger than the closure of \(T(\Gamma)\), even if \(\Gamma\) is restricted to the class of purely hyperbolic groups of the second kind. For a finitely generated Fuchsian group \(\Gamma\) it was proved by Zhuravlev that \(T(\Gamma)\) is the connected component of \(0\) in \(\text{Int}(S(\Gamma))\). The proof given here follows a method developed by F. Gehring but it requires a very delicate analysis for its execution.
scientific article; zbMATH DE number 871817