Integrable Teichmüller space

Parametrization of the $p$-Weil-Petersson curves: holomorphic dependence

“…holomorphic as is shown in[39, Theorem 2.4]. Moreover, as the composition α • coincides with the Bers projection σ : M p (L) → T p given by µ → S fµ , there is a local holomorphic inverse of α • at any point of T p (see[39, Theorem 2.1] and[45, Proposition 4.3]).…”

“…holomorphic as is shown in[39, Theorem 2.4]. Moreover, as the composition α • coincides with the Bers projection σ : M p (L) → T p given by µ → S fµ , there is a local holomorphic inverse of α • at any point of T p (see[39, Theorem 2.1] and[45, Proposition 4.3]). It follows from this property that α also has a local holomorphic inverse at any point of T p , which shows that α −1 is holomorphic.…”

“…The p-Weil-Petersson metric has been introduced in [21,22]. Moreover, Tang and Shen [39] characterized the p-Weil-Petersson class W p of quasisymmetric homeomorphisms of R having quasiconformal extensions to U with complex dilatation in M p (U) by the p-Besov space B R p (R) of realvalued functions, which coincides with H 1/2 R (R) for p = 2. These generalizations are usually straightforward.…”

The $p$-Weil-Petersson Teichmüller space and the quasiconformal extension of curves

“…For p > 2, the normalized p-Weil-Petersson class W p (R) can be defined similarly by just changing 2-integrability into p-integrability. The generalization from the case of p = 2 to the general p ≥ 2 is natural, and several works have been done in this direction (see [13,16,17,26,33]). These generalizations are usually straightforward, but there are really a few crucial differences in the arguments between the cases of p = 2 and p > 2.…”

“…Taking the space M p (U) of the Beltrami coefficients that are p-integrable with respect to the hyperbolic metric on U for p ≥ 2, the p-Weil-Petersson Teichmüller space T p (U) is given by the Teichmüller projection π : M p (U) → T p (U). It is known that T p (U) has a unique complex Banach manifold structure via the Bers embedding through the Schwarzian derivative (or via the logarithmic derivative embedding) such that the Teichmüller projection π is holomorphic with local holomorphic inverse (see [26,31]). This can be considered on the lower halfplane L in the same way.…”