The Loewner differential equation and slit mappings. (English) Zbl 1078.30005

Marshall, Donald E.; Rohde, Steffen

J. Am. Math. Soc. 18, No. 4, 763-778 (2005).

The authors study the solutions of the Loewner differential equation \[ \partial_t F(z,t)=z\cdot p(z,t) \partial_z f(z,t), \] \(p(z,t)\) is analytic in \(z\) and has positive real part, which describes the evolution of a normalized Loewner chain \(f_t(z): D\rightarrow\Omega_t,\) \(D=\{z:| z| <1\}.\) More precisely, they investigate the geometry of the solutions in case of slit domains \(\Omega_t=\Omega\setminus \gamma[t,t_1],\) where \(\Omega\subseteq C\) is simply connected and \(\gamma\) parametrizes a simple arc that is contained in \(\Omega\) except for one end point \(\gamma(t_1)\in \partial\Omega.\) In this case the Loewner equation becomes \[ \partial_t f=z\frac{\lambda(t)+z}{\lambda(t)-z}\partial_z t, \] where \(\lambda(t)=f_t^{-1}(\gamma(t))\in \partial D.\) The main result of this paper is a sharp condition on \(\lambda\) that guarantees slit domains.

Reviewer: Nikola Tuneski (Skopje)

MSC:

30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C20	Conformal mappings of special domains
30C62	Quasiconformal mappings in the complex plane
30C30	Schwarz-Christoffel-type mappings

Keywords:

conformal maps; harmonic measure; quasiconformal maps; quasiarc; conformal welding; Loewner’s differential equation; Lipschitz continuous; Hölder continuous

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