Complex geodesics in convex tube domains (Q2792164)

Let \(D\subset\mathbb C^n\) be a taut convex tube domain. The author studies conditions under which a holomorphic mapping \(\varphi=(\varphi_1,\dots,\varphi_n):\mathbb D\longrightarrow D\) is a complex geodesic (i.e., there exists a holomorphic mapping \(f:D\longrightarrow\mathbb D\) such that \(f\circ\varphi\) is the identity of the unit disc \(\mathbb D\)). Let \(\mu=(\mu_1,\dots,\mu_n)\) be the system of boundary measures for \(\varphi\), i.e., \(\mu_j\) is a finite real Borel measure on \(\mathbb T:=\partial\mathbb D\) such that NEWLINENEWLINE\[NEWLINE\varphi_j(\lambda)=\frac1{2\pi}\int_{\mathbb T}\frac{\zeta+\lambda}{\zeta-\lambda}d\mu_j(\zeta)+i\text{Im}\varphi(0), \quad\lambda\in\mathbb D,\quad j=1,\dots,n.NEWLINE\]NEWLINENEWLINE The main result of the paper states that \(\varphi\) is a complex geodesic iff there exist \(a\in\mathbb C^n\), \(b\in\mathbb R^n\) such that the mapping \(\displaystyle h(\lambda):=\overline a\lambda^2+b\lambda+a:\mathbb C\longrightarrow\mathbb C^n\) is not trivial and the measure \(\displaystyle\sum_{j=1}^n\overline\lambda h_j(\lambda)(\text{Re}z_jd\mathcal L^{\mathbb T}(\lambda)-\mu_j(\lambda))\) is negative for every \(z\in D\), where \(\mathcal L^{\mathbb T}\) denotes the Lebesgue measure on \(\mathbb T\).NEWLINENEWLINEAs an application of the above result the author presents examples of taut convex tube domains for which complex geodesics can be effectively calculated.