Quasiconformal contactomorphisms and polynomial hulls with convex fibers

Consider the polynomial hull of a smoothly varying family of strictly conve x smooth domains fibered over the unit circle. It is well-known that the bo undary of the hull is foliated by graphs of analytic discs. We prove that t his foliation is smooth, and we show that it induces a complex flow of cont actomorphisms. These mappings are quasiconformal in the sense of Koranyi an d Reimann. A similar bound on their quasiconformal distortion holds as in t he one-dimensional case of holomorphic motions. The special case when the f ibers are rotations of a fixed domain in C-2 is Studied in details.