The Grunsky Coefficients as a Model of Universal Teichmüller Space

Lemma 1. [7] Let D∗be a quasidisk containing z = ∞. Then for every function f ∈Σ0(D∗), its Grunsky and Teichmüller norms are related by κ(f) ≤k(f) ≤κ(f)/αD(f). (8) Note that the right inequality in (8) is given in [7] for functions f having a unique extremal extension. Since, due to [4], the Schwarzians Sf of such functions fill a dense open set in the space T and both Teichmüller and Grunsky norms are continuous on this space, Lemma 1 holds in its general form presented above. Advances in Analys

Lemma 2. Let E, T be open subsets of complex Banach spaces X, Y and B(E) be a Banach space of holomorphic functions on E with sup-norm. If ϕ(x, t) is a bounded map E × T →B(E) such that t 7→ϕ(x, t) is holomorphic for each x ∈E, then the map ϕ is holomorphic. Note that holomorphy of ϕ(x, t) in t for fixed x implies the existence of complex directional derivatives ϕ′ t(x, t) = lim ζ→0 ϕ(x, t + ζv) −ϕ(x, t) ζ = 1 2πi Z |ξ|=1 ϕ(x, t + ξv)

Lemma 2 implies that this Schwarzian is also holomorphic as an element of the space B. This provides the holomorphy of the inverse map χ−1. We have established that the map χ is a biholomorphis homeomorphism between the domain T in B and the open set T∞= χ(T) ⊂L. Thus the topological properties of T such as path-wise connectedness and contractibility are carried to T∞. This completes the proof of the theorem. 4 Additional Remarks 1. The arguments used in the proof of the theorem are valid only f