Stream lines, quasilines and holomorphic motions

1 Background The theory of holomorphic motions, introduced by Mané-Sad-Sullivan [1] and advanced by Slodkowski [2], has had a significant impact on the theory of quasiconformal mappings. A reasonably thorough account of this is given in our book [3]. In [4, 5] we established some classical distortion theorems for quasiconformal mappings and used the theory to develop connections between Schottky’s theorem and Teichmüller’s theorem. We also gave sharp estimates on the distortion of quasicircles which in turn gave estimates for the distortion of extensions of analytic germs as studied in [6]. Here we consider the geometry of stream lines for ideal fluid flow in a domain and establish bounds on their distortion in terms of a reference line. These bounds come from an analysis of the geometry of the level lines of the hyperbolic metric and seem to be of independent interest. When the reference line is known to be a quasiline—the image of R under a quasiconformal map of C—which occurs for instance when there is some symmetry about, it follows that all level lines are quasilines and it is possible to give explicit distortion estimates which contains global geometric information—such as bounded turning—for the curve, see for instance (8) below. As such these estimates will have implications for parabolic linearisations. We first recall the two basic notions we need here.