Holomorphic families of Riemann mapping functions
This paper concerns some applications of the \(\lambda\)-lemma of \textit{R. Mañé}, \textit{P. Sad} and \textit{D. Sullivan} [Ann. Sci. Éc. Norm. Supér. IV. Sér. 16, 193-217 (1983; Zbl 0524.58025)], see also \textit{L. Bers} and \textit{H. Royden} [Acta Math. 157, 259-286 (1986)]. In particular, the authors investigate the holomorphic dependence on \(\lambda\) of suitably normalised Riemann mapping functions \(f_{\lambda}\) of the unit disc D to the region inside \(\gamma_{\lambda}\), where \(\gamma_{\lambda}\) is a holomorphically varying Jordan curve in \({\mathbb{C}}\), and \(\lambda\in D\). If, in this situation, the Riemann mapping functions from D to the interior and exterior of \(\gamma_{\lambda}\) depend holomorphically on \(\lambda\), and if \(\gamma_ 0\) has measure zero, then the motion of \(\gamma_ 0\) is ''trivial'', i.e., of the form \[ F(\lambda,z)=(a(\lambda)z+b(\lambda))/(c(\lambda)z+d(\lambda)), \] where a,b,c,d are holomorphic in D. A normalisation of this form is possible if and only if for each \(\lambda\), the function \(F(\lambda,\cdot): \gamma_ 0\to \gamma_{\lambda}\) is the boundary value of a function holomorphic in \(G_ 0\), the region inside \(\gamma_ 0\). For example, if \(\gamma_ 0\) has a continuously turning unit normal \(\bar n(z)\), then the holomorphic motion \(F(\lambda,z)=z+\lambda \bar n(z)\) (the Hadamard variation) has a holomorphic Riemann mapping function if and only if \(\gamma_ 0\) is a circle. A further application is a result of Sullivan that if \(R_{\lambda}\) is a rational function with Siegel disc \(G_{\lambda}\) centred at a(\(\lambda)\), and with boundary determined (in a precise way) by critical points \(c_ i(\lambda)\), \(1\leq i\leq n\), so that \(a(\lambda)\), \(c_ i(\lambda)\) vary holomorphically for \(\lambda\) in some simply connected region \(\Omega\), then \(\partial G_{\lambda}\) and the Riemann mapping functions to \(G_{\lambda}\) vary holomorphically with \(\lambda\).
scientific article; zbMATH DE number 3981490