HOLOMORPHIC MOTIONS AND TEICHMULLER-SPACES

We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane C extends to a holomorphic motion of C . As a consequence we prove that every holomorphic map of the unit disc into Teichmuller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmuller metric.