Perturbation of the tangential slit by conformal maps
Suppose that, for any \(T>0\), \(\gamma:[0,T]\to\mathbb H\cup\mathbb R\) is a simple curve, \(\gamma(0)\in\mathbb R\), \(\gamma(0,T]\subset\mathbb H=\{z\in\mathbb C:\text{Im}\,z>0\}\). There is a unique conformal map \(g_t\) from \(\mathbb H_t=\mathbb H\setminus\gamma[0,t]\), \(t\in[0,T]\), onto \(\mathbb H\) with the hydrodynamic normalization \[ g_t(z)=z+\frac{2t}{z}+O\left(\frac{1}{z^2}\right),\;\;\;z\to\infty, \] and \(g_t\) obeys the chordal Loewner differential equation \[ \frac{\partial g_t(z)}{\partial t}=\frac{2}{g_t(z)-\lambda(t)},\;\;\;g_0(z)=z,\;\;\;z\in\mathbb H, \] with a continuous real-valued driving function \(\lambda\). We say that the trace \(\gamma\) is generated by \(\lambda\). In particular, the circular arc \(\Gamma:=\{i+e^{i\theta}:-\frac{\pi}{2}\leq\theta\leq0\}\) is generated by a Hölder continuous driving function with exponent 1/3. The authors consider the behavior of a driving function \(\lambda\) which generates \(\varphi(\Gamma)\), where \(\varphi\) is an analytic function defined in a neighborhood of the origin, \[ \varphi(w)=\sum_{n=1}^{\infty}a_nw^n,\;\;\;a_1>0,\;\;\;a_n\in\mathbb R,\;\;\;n\geq2. \] The main result of the paper is given in the following theorem. Theorem 1.1: Let \(\varphi(\Gamma)\) be a trace generated by the chordal Loewner differential equation. Then its driving function \(\lambda\) is of the form \[ \lambda(t)=(12\pi\varphi'(0)t)^{1/3}+o(t^{1/3}),\;\;t\to0. \] In the special case \(\varphi^{-1}(z)=P(z)=z\prod_{k=1}^n(1-c_kz^k)\), \(c_k\in\mathbb R\), \(k=1,\dots,n\), there is a neighborhood \(D\) of the origin such that \(P\) is univalent in \(D\). Assume that \(\Gamma(t)\) is a parametrization of \(\Gamma\) by \(t\) and an arc \(\Gamma[0,t_0]\) of \(\Gamma\) is contained in \(P(D)\). The authors deduce an asymptotic expansion for \(\lambda(t)\) near \(t=0\) when \(\lambda\) generates \(P^{-1}(\Gamma[0,t_0])\).
scientific article; zbMATH DE number 6877197
scientific article