Conformal welding for domains close to a disk
This interesting paper deals with conformal welding of starlike Jordan domains. Let \(D\) be a Jordan domain containing the origin and let \(f:{\mathbb D}\to D\) and \(F:{\mathbb D^*}\to\overline{\mathbb C}\setminus\overline D\) be the conformal mappings of the unit disk~\({\mathbb D}\) onto~\(D\) and of the exterior \({\mathbb D^*}:=\overline{\mathbb C}\setminus\overline{\mathbb D}\) of the unit disk onto the exterior of~\(D\), respectively, subject to the normalization \(f(0)=0\), \(f'(0)>0\), \(F(\infty)=\infty\), \(F'(\infty)>0\). Being extended by continuity to the unit circle these maps define a homeomorphism \(\gamma:\partial{\mathbb D}\to \partial{\mathbb D};\, e^{is}\mapsto e^{i\sigma}\) by the equation \(F(\gamma(e^{is}))=f(e^{is})\) for all \(s\in[0,2\pi]\). The homeomorphism \(\gamma\) is said to be the conformal welding associated to the Jordan domain~\(D\). The author studies conformal welding associated to starlike domains close to the unit disk. Let \(D\) be given in polar coordinate by \[ D:=\big\{\rho e^{i\psi}:\,\psi\in[0,2\pi],\,0\leq \rho\leq 1-\delta(\psi)\big\}, \] where \(\delta:{\mathbb R}\to(-\infty,1)\) is a twice differentiable \(2\pi\)-periodic function. Using a variational formula due to \textit{G. V. Siryk} [Usp. Mat. Nauk 11, No. 5(71), 57--60 (1956; Zbl 0074.06102)] and its analogue for the ``exterior'' map~\(F\) (Theorem~2), the author establishes the following asymptotic relation (Proposition 1) for the conformal welding \({e^{i\sigma}=\gamma(e^{is})}\), \(0\leq s\leq 2\pi\), associated to~\(D\): \[ s+h(s)=\sigma-h(\sigma)+ O(\epsilon^2)\text{ for all \(s\in[0,2\pi]\)} \] \[ \text{as }\epsilon:=\sup\{|\delta^{(k)}(\psi)|:\psi\in[0,2\pi],\,k=0,1,2\}\to+0, \] where \[ h(x):=\int_0^{2\pi}(\delta(\psi)-\delta(x))\cot\frac{\psi-x}{2}\,d\psi,\quad 0\leq x\leq2\pi. \] Similar formulas are obtained for domains \(D(\varphi,\alpha)\) bounded by \(l_1:=\big\{e^{i\theta}:\theta\in[\varphi,2\pi-\varphi]\big\}\) and the circular arc \(l_2\subset\overline{\mathbb D}\) with the end-points at \(e^{\pm i\varphi}\) forming angle \(\alpha\pi\) with~\(l_1\). Two situations are discussed: in Proposition 2, \(\varphi\in(0,\pi)\) is fixed and \(1-\alpha\to+0\), while in Proposition 3, \(\alpha\in[\frac12,1)\) is fixed and \(\varphi\to+0\). In Theorem 3, the author obtains an analogous variational formula for conformal weldings associated to families of starlike circular quadrangles with fixed inner angles. As a byproduct, it is proved (Theorem 1) that the conformal mappings \(w(t)=f_t(z;1/2)=e^{-t}z+a_2(t)z^2+\ldots\), \(t\geq0\), of \({\mathbb D}\) onto \(D(\varphi_t,1/2)\) (where for each \(t\geq0\) the parameter \(\varphi_t\in(0,\pi)\) is to be appropriately chosen) satisfy the Loewner-type equation for typically real univalent functions \[ \frac{dw}{dt}=-w\frac{1-w^2}{1-2\cos u(t)w+w^2},\quad w|_{t=0}=z, \] with \(u(t)=\arccos e^{-t}\). This contributes to explicit Kufarev type examples of non-slit solutions to the Loewner equation.
scientific article; zbMATH DE number 6112464