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Quasiconformal homogeneity of genus zero surfaces. (English) Zbl 1223.30006

A Riemann surface \(M\) is called \(K\)-quasiconformally homogeneous if for every two points \(p,q \in M\), there is a \(K\)-quasiconformal homeomorphism \(f:M \to M\) such that \(f(p)=q\). This notion was introduced by F. W. Gehring and B. P. Palka [“Quasiconformally homogeneous domains”, J. Anal. Math. 30, 172–199 (1976; Zbl 0349.30019)] in the setting of genus zero surfaces.
The paper under review answers a question by Gehring and Palka by proving the existence of a constant \(\mathcal{K} >1\) such that if \(M\) is a \(K\)-quasiconformal homogeneous hyperbolic genus zero surface that is not the unit disk \(\mathbb{D}\), then \(K \geq \mathcal{K}\).
Using a result from Gehring and Palka on transitive conformal families, this implies that if \(M\) is a \(K\)-quasiconformal homogeneous genus zero surface for \(K<\mathcal{K}\), then \(M\) is conformally equivalent to either \(\mathbb{P}^1\), \(\mathbb{C}\), \(\mathbb{C} \setminus \{ 0 \}\) or \(\mathbb{D}\).
The analagous problem in higher dimensions has been solved by P. Bonfert-Taylor, R. D. Canary, G. Martin and E. Taylor [“Quasiconformal homogeneity of hyperbolic manifolds”, Math. Ann. 331, No. 2, 281–295 (2005; Zbl 1063.30020)].

MSC:

30C62 Quasiconformal mappings in the complex plane
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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References:

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