Evolution of conformal maps with quasiconformal extensions

We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M=DiffS1/RotS1 embedded into T. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. This approach allows us to understand the Laplacian growth (Hele-Shaw problem) as a flow in the Teichmüller space.