Abstract
Douady–Earle extension provides a quasiconformal extension of a quasisymmetric homeomorphism to the unit disk and induces a continuous self-map σ of Beltrami coefficients. In this note, we show that σ is also continuous (under a stronger topology) on those Beltrami coefficients which induce vanishing Carleson measures and thus project to the VMO-Teichmüller space. An immediate consequence is that the VMO-Teichmüller space is contractible.
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