2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/98819We study one-parameter curves on the universal Teichmüller space $T$ and on the homogeneous space $M=\Diff S^1/\Rot S^1$ 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. Some applications to Hele-Shaw flows of viscous fluids are given.32 pages, 3 figuresAnalysis of PDEsComplex Variables30C35, 22E65, 30C62, 30F60, 76D27text