Kobayashi's and Teichmüller's metrics and Bers complex manifold structure on circle diffeomorphisms (Q1987564)

For \(0<\alpha \leq 1\), denote by \(\mathcal{C}^{1+\alpha}\) the set of orientation-preserving diffeomorphisms of the unit circle \(S^1\) whose derivative is \(\alpha\)-Hölder continuous. Then the corresponding Teichmüller space \(\mathcal{T} \mathcal{C}^{1+\alpha}\) can be identified with the subset of \(\mathcal{C}^{1+\alpha}\) which fixes \(1,-1,i\). Moreover, denote by \(\mathcal{T} \mathcal{C}^{1+H}\) the union \(\bigcup_{0<\alpha \leq 1} \mathcal{T} \mathcal{C}^{1+\alpha}\). After an extensive introduction to the required material, the first main result of the paper is Theorem 4.2, which states that \(\mathcal{T} \mathcal{C}^{1+H}\) can be equipped with a Bers complex manifold structure. This can be viewed as a smooth version of the analogous result for universal Teichmüller spaces. Moreover, this is the largest subspace in the space of all \(C^1\) circle diffeomorphisms on which this structure can be given. The author then proves in Theorem 5.1 that the Kobayashi and Teichmüller metrics coincide, again making a parallel with well-known results in quasiconformal Teichmüller theory. The proofs of the results use well-known techniques in Teichmüller theory, such as Schwarzian derivatives, quasiconformal extensions of quasisymmetric maps, and computations involving Beltrami differentials.