A holomorphic motion is a family of injections of a subset of the Riemann sphere, indexed by a complex parameter in the unit disk, that begins at the identity and varies holomorphically in the parameter. A student learns why such a family is far more rigid than it looks: it always extends to a motion of the entire sphere, every map in it is automatically quasiconformal with dilatation controlled by the parameter, and its Beltrami coefficient depends holomorphically on that parameter. The extension is proved by a fixed-point argument for a compact operator built from the Cauchy kernel, and the resulting continuity estimates make holomorphic motions a tool for transferring hyperbolic-metric bounds into Holder estimates, for comparing metrics on Teichmuller space, and for linearizing parabolic germs in complex dynamics.
A holomorphic motion of a set E in the Riemann sphere is a family of maps h(c,z), indexed by a complex parameter c in the unit disk, that starts at the identity when c=0, is injective in z for each f…