Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

http://dx.doi.org/10.4153/CJM-1999-040-3
Canad. J. Math. 51(1999), 915-935

  Published:1999-10-01
 Printed: Oct 1999

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Kor\'anyi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in $\C^2$ is studied in details.