Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces

Introduction of a complex structure on the p-integrable Teichmüller space

“…Specifically, we characterize each point of T p (Γ) by its Douady-Earle extension, which is a quasiconformal self-mapping on ∆ with conformal naturality (see [6]). Originally, Cui [5] proved this result in the case of T 2 (1) and Tang [20] extended it to T p (1) for p ≥ 2, where 1 = {id ∆ } is the trivial group. In the proof, they applied the Dirichlet integral of harmonic self-maps on ∆ obtained by the Poisson integral (see [4]).…”

Holomorphic contractibility and other properties of the Weil-Petersson and VMOA Teichmüller spaces

“…It was shown respectively in [5], [3] and [27] The Kobayashi pseudo metric is defined on any complex Banach manifold, and hence on any Teichmüller space. Then one likes to study whether or not the Kobayashi pseudo metric and the Teichmüller metric coincide with each other on a Teichmüller space.…”

“…which is called the integrable Teichmüller space in [3] or the Weil-Petersson Teichmüller space in [22]…”

“…Furthermore, T 2 (D) is a group under the composition [3]. For more studies of T 2 (D), we refer to [16], [24] and [26].…”

Dirichlet problem and Sokhotski-Plemelj jump formula on Weil-Petersson class quasidisks

“…The WP-class quasicircles are a strict subclass of the so-called asymptotically conformal quasicircles, which have been an object of interest for the last decade. Their study was initiated by Cui [5] and Guo [9], in connection with finding a theory of the universal Teichmüller space based on L p Beltrami differentials and conformal maps. The memoir [18] of Takhtajan and Teo obtained wide-ranging results on the WP-class universal Teichmüller space, including for example sewing formulas for the Laplacian and potentials for the Weil-Petersson metric.…”