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Isometries of the Teichmüller metric. (English) Zbl 0933.32027

The aim of the paper is to apply the complex Finsler geometry (finite and infinite) to the study of Teichmüller spaces in order to characterize these spaces in terms of isometries of intrinsic metrics. The main result is
Theorem. A taut connected complex manifold \(N\) is biholomorphic to a finite dimensional Teichmüller space \(T(\Gamma)\) if and only if there exists a holomorphic map \(F: N \to T(\Gamma)\) which is an isometry for the Kobayashi metric at one point.
The paper is organized as follows. First, there is given a short overview of complex Finsler metrics on complex manifolds of finite and infinite dimensions. Then, on the unit ball \(M\) in \(L^\infty(H^+,C)\), where \(H^+\) is the upper half plane in \(C\), the Teichmüller metric \(\sigma(\mu; \nu)=||{|\nu|\over 1-|\mu|^2}||_\infty\) is considered as a Finsler metric, and in a direct way there is proved that, for a Fuchsian group \(\Gamma\), the Teichmüller, Carathéodory and Kobayashi metrics (distances) on \(M(\Gamma) = M \cap L^\infty(\Gamma)\) coincide.
The holomorphic curvature of the Kobayashi-Teichmüller metric on the Teichmüller space \(T(\Gamma)\) is proved to be a constant equal to \(-4\). With the use of this fact, the properties of infinitesimal complex geodesics are described (for a complex Finsler manifold \((M,F)\), an infinitesimal complex geodesic is a map \(\phi : \Delta \to M\) such that at each point \(\zeta_0 \in \Delta\) there holds \(F(\phi(\zeta_0), \phi'(\zeta_0)) = {1 \over 1-|\zeta_0|^2}\)). For example, there exists an infinitesimal complex geodesic passing through a given point in a given direction, and if \(T(\Gamma)\) is finite-dimensional, this geodesic is uniquely determined.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
32F45 Invariant metrics and pseudodistances in several complex variables
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