The dilatation function of a holomorphic isotopy

Krushkal, Samuel L.

Funct. Approx. Comment. Math., Tome 40 (2009) no. 1, p. 75-90 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2009-03-15
Classification:  Univalent function,  quasiconformal map,  dilatation,  subharmonic function,  universal Teichmüller space,  hyperbolic metrics,  pluricomplex Green function,  30C55,  30C62,  30F45,  30F60,  32U55