Earle, Clifford J. The holomorphic contractibility of two generalized Teichmüller spaces. (English) Zbl 1150.30376 Publ. Inst. Math., Nouv. Sér. 75(89), 109-117 (2004). Let \(R\) be a hyperbolic Riemann surface. The Teichmüller space \(T(R)\) of \(R\) is a quotient space of the set of all quasiconformal mappings whose domain is \(R\). Let \(T_0(R)\) denote the closed complex submanifold of \(T(R)\) which consists of the equivalence classes of asymptotically conformal quasiconformal mappings with domain \(R\). Both spaces \(T(R)\) and \(T_0(R)\) are known to be contractible. The author then asks whether they are holomorphically contractible. Here, a complex manifold \(X\) is said to be holomorphically contractible to a point \(x_0\in X\) if there is a continuous mapping \(F:[0,1]\times X\to X\) such that (i) \(F(0,x)=x\) and \(F(1,x)=x_0\) for all \(x\in X\) and (ii) \(F(t,x)\) is holomorphic and fixes \(x_0\) for all \(t\in[0,1]\). Let \(\Delta\) be the unit disk and set \(\Delta'=\Delta\smallsetminus\{0\}\). The author proves by explicit constructions that the spaces \(T_0(\Delta)\) and \(T_0(\Delta')\) are holomorphically contractible to their fixed points. Reviewer: David Kalaj (Podgorica) Cited in 3 Documents MSC: 30F60 Teichmüller theory for Riemann surfaces 46G20 Infinite-dimensional holomorphy Keywords:Teichmüller space; Rieman surface; Holomorphic map PDFBibTeX XMLCite \textit{C. J. Earle}, Publ. Inst. Math., Nouv. Sér. 75(89), 109--117 (2004; Zbl 1150.30376) Full Text: DOI EuDML