Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Abstract

Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with $\alpha$-Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local $C^{1+\alpha}$-topology at the base point.