Cross-ratio distortion and Douady-Earle extension: II. Quasiconformality and asymptotic conformality are local |
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Authors: | Jun Hu Oleg Muzician |
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Institution: | 1.Department of Mathematics,Brooklyn College of Cuny,Brooklyn,USA;2.Graduate Center of Cuny,New York,USA;3.Department of Mathematics,Cuny Graduate Center,New York,USA |
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Abstract: | Let f be an orientation-preserving circle homeomorphism and Φ the Douady-Earle extension of f. In this paper, we show that the quasiconformality and asymptotic conformality of Φ are local properties; i.e., if f is quasisymmetric or symmetric on an arc of the unit circle, then Φ is quasiconformal or asymptotically conformal nearby.
Furthermore, our methods enable us to conclude the global quasiconformality and asymptotic conformality from local properties.
In the quasiconformal case, our methods also enable us to provide an upper bound for the maximal dilatation of Φ on a neighborhood
of the arc in the open unit disk in terms of the cross-ratio distortion norm of f on the arc. |
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