Functiones et Approximatio Commentarii Mathematici

The dilatation function of a holomorphic isotopy

Samuel L. Krushkal

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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.

Article information

Funct. Approx. Comment. Math., Volume 40, Number 1 (2009), 75-90.

First available in Project Euclid: 30 March 2009

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Zentralblatt MATH identifier

Primary: 30C55: General theory of univalent and multivalent functions 30C62: Quasiconformal mappings in the plane 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Secondary: 30F60: Teichmüller theory [See also 32G15] 32U55

Univalent function quasiconformal map dilatation subharmonic function universal Teichmüller space hyperbolic metrics pluricomplex Green function


Krushkal, Samuel L. The dilatation function of a holomorphic isotopy. Funct. Approx. Comment. Math. 40 (2009), no. 1, 75--90. doi:10.7169/facm/1238418799.

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