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The holomorphic contractibility of two generalized Teichmüller spaces. (English) Zbl 1150.30376

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.

MSC:

30F60 Teichmüller theory for Riemann surfaces
46G20 Infinite-dimensional holomorphy
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