Lambda-Lemma is an important tool in holomorphic dynamics. It states that a holomorphic motion of a subset of the complex plane can be extended to the holomorphic motion of the whole plane. In general holomorphic motions in higher dimensions do not have any good properties. The situation changes if one restricts the motion.
Let S_lambda be a family of Riemann surfaces.We assume that S_lambda are conformally equivalent to disks with n holes and that their boundaries move holomorphically in C^2. Moreover, we assume that the motion of the boundary is quasisymmetric. Then it can be extended to the holomorphic motion of the Riemann surfaces.