Quasiconformal variation of slit domains (Q2731933)

Let \(\Omega \subset \mathbb C\) be a simply connected domain with \(0 \in \Omega\), and let \(f \colon [0,T) \to \Omega\), \(0 < T \leq \infty\) be a parametrization of a Jordan arc \(\Gamma\) in \(\Omega\). It is assumed that \(f\) is regular, i.e., \(f'\) exists and has no zeros, and that \(\Gamma\) is a closed subset of \(\Omega\). Then \(f(t)\) approaches the boundary of \(\Omega\) as \(t \to T\), and for all \(t \in [0,T)\) the arc \(\Gamma_t = f([t,T))\) is a closed subset of \(\Omega\) so that the domain \(\Omega_t = \Omega \setminus \Gamma_t\) is simply connected. Finally, let \(0 \notin \Gamma\) which means \(0 \in \Omega_t\) for all \(t \in [0,T)\). Then, denote by \(R(t)\) the conformal radius of \(\Omega_t\) with respect to \(0\). NEWLINENEWLINENEWLINEThe first result states that if \(f\) is real analytic on \([0,T)\), then \(R\) is a real analytic function of \(t\). This answers a question of D.~Gaier. Next, let \(z \mapsto h(z,t)\) be the conformal map of the open unit disk \(\Delta\) onto \(\Omega_t\) such that \(h(0,t)=0\) and \(h(1,t)=f(t)\). If \(f\) is \(C^n\) on \([0,T)\) with \(n \geq 2\), then \((z,t) \mapsto h(z,t)\) is a \(C^{n-1}\) function on \(\Delta \times [0,T)\). In addition, there exist real valued \(C^{n-2}\) functions \(\alpha\) and \(\beta\) on \([0,T)\) such that \(\alpha(t)>0\) and NEWLINE\[NEWLINE {\partial h \over \partial t} (z,t) = z{\partial h \over \partial z} (z,t) \left[\alpha(t){1+z \over 1-z}+i\beta(t) \right], \quad z \in \Delta, \;t \in [0,T) . \tag{1} NEWLINE\]NEWLINE If \(f\) is real analytic on \([0,T)\), then so are the map \((z,t) \mapsto h(z,t)\) and the function \(\alpha\) and \(\beta\). By changing the regular parametrization \(f\) and the normalization of the conformal maps, equation (1) can be put into a classical Löwner form. Then it is shown that if \(f\) is \(C^n\), \(n \geq 2\), then the standard parametrization of \(\Gamma\) is regular. Also the standard parametrization and the Löwner \(\kappa\)-function are both \(C^{n-2}\). They are real analytic if \(f\) is real analytic. Finally, the situation where \(f\) is slightly less than \(C^2\) is discussed. NEWLINENEWLINENEWLINEThe method of proof is based on quasiconformal variations. The proof of the first result uses a theorem of B.~Rodin [Complex Variables, Theory Appl. 5, 189-195 (1986; Zbl 0597.30013)] about the variation of the conformal map when the boundary of a simply connected domain undergoes a holomorphic motion. Rodin's theorem is proved by combining the theory of holomorphic motions with a method of quasiconformal variation introduced by \textit{L. V. Ahlfors} [Ann. Acad. Sci. Fenn., Ser. A I 206, 1-22 (1955; Zbl 0067.30702)]. Ahlfors's method can be applied directly to both the real analytic and \(C^n\) case, and it is used in the proof of the second result.