On Douady–Earle extension and the contractibility of the VMO-Teichmüller space

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.