The dilatation function of a holomorphic isotopy

Abstract

Every nonvanishing univalent function $f(z)$ in the disk $\Delta^* = \widehat{\mathbb{C}} \setminus \overline{\Delta}, \Delta = \{|z| < 1\}$, for example, with hydrodynamical normalization, generates a complex isotopy $f_t (z) = t f(t^{-1} z): \Delta^* \times \Delta \to \widehat{\mathbb{C}}$, which is a special case of holomorphic motions and plays an important role in many topics. Let $q_f$ denote the minimal dilatation among quasiconformal extensions of $f$ to $\widehat{\mathbb{C}}$. In 1995, R. Kühnau raised the questions whether the dilatation function $q_f(r) = q_{f_r}$ is real analytic and whether the function $f$ can be reconstructed if $q_f(r)$ is given. The analyticity of $q_f$ was known only for ellipses and for the Cassini ovals. Our main theorem provides a wide class of maps with analytic dilatations and implies also a negative answer to the second question.