Remarks on the metric induced by the Robin function (Q2909229)

\textit{N.~Levenberg} and \textit{H. Yamaguchi} [Robin functions for complex manifolds and applications. Mem. Am. Math. Soc. 448, 156 p. (1991; Zbl 0742.31003)] introduced the Kähler metric (the so-called \(\Lambda\)-metric) NEWLINENEWLINE\[NEWLINE ds_z^2=\sum_{\alpha,\beta=1}^n\frac{\partial^2\log(-\Lambda)}{\partial z_{\alpha}\partial\bar z_{\beta}}(z)dz_{\alpha}\otimes d\bar z_{\beta} NEWLINE\]NEWLINE induced by the Robin function NEWLINENEWLINE\[NEWLINE \Lambda_D(p):=\lim_{z\to p}\big(G_D(z,p)-|z-p|^{-2n+2}\big),\quad p\in D, NEWLINE\]NEWLINE NEWLINEwhere \(G_D(\cdot,p)\) is the Green function for a smoothly bounded domain \(D\subset\mathbb C^n\) with pole at \(p\). NEWLINENEWLINENEWLINE In the paper under review the authors study further properties of this metric.NEWLINENEWLINELet \(D\subset\mathbb C^n\) be a bounded domain. They show the following theorems. NEWLINENEWLINENEWLINE 1. Let \(D\) be regular and let \(\Gamma\subset\partial D\) be a \(\mathcal C^2\)-smooth open piece, \(x_0\in\Gamma\). Suppose that \(\psi\) is local defining function for \(\Gamma\) near \(z_0\) and let NEWLINE\[NEWLINE H=\left\{z\in\mathbb C^n:2\Re\left(\sum_{\alpha=1}^n\frac{\partial\psi}{\partial z_{\alpha}}(z_0)z_{\alpha}\right)-1<0\right\}. NEWLINE\]NEWLINE Then for all \(A,B\in\mathbb N_0^n\) NEWLINENEWLINE\[NEWLINE \lim_{z\to z_0}(-1)^{|A|+|B|}\big(\mathcal D^{A,\bar B}\Lambda_D(z)\big)\big(\psi(z)\big)^{2n-2+|A|+|B|}=\mathcal D^{A,\bar B}\Lambda_H(0). NEWLINE\]NEWLINE NEWLINE 2. Let \(D\) be \(\mathcal C^{\infty}\)-smooth pseudoconvex, \(z_0\in\partial D\). Suppose that \(\psi\) is a defining function for \(D\). Then NEWLINENEWLINE\[NEWLINE \lim_{z\to z_0}\Big(ds_z^2\big(v_N(z),v_N(z)\big)\Big)^{1/2}\big(-\psi(z)\big)=(2n-2)^{1/2}|v_N(z)||\partial\psi(z_0)| NEWLINE\]NEWLINE NEWLINEand NEWLINENEWLINE\[NEWLINE\begin{multlined} \lim_{z\to z_0}\Big(ds_z^2\big(v_H(z),v_H(z)\big)\Big)^{1/2}\big(-\psi(z)\big)^{1/2}=\\ (2n-2)^{1/2}|v_N(z)||\partial\psi(z_0)|\big\langle\mathcal L_{\psi}(z_0)v_H(z_0),v_H(z_0)\big\rangle^{1/2},\end{multlined} NEWLINE\]NEWLINE NEWLINEwhere the limits are locally uniform in \(v\), \(v=v_H(p)+v_N(p)\in H_p(\partial D)\oplus N_p(\partial D)\) is a canonical splitting along the complex tangential and normal directions at \(p\in\partial D\).NEWLINENEWLINE3. Let \(D\) be \(\mathcal C^{\infty}\) smooth strongly pseudoconvex. Then there is a constant \(C_A\geq1\) such that NEWLINE\[NEWLINE C_A^{-1}d_A\leq d_R\leq C_Ad_A, NEWLINE\]NEWLINE where \(A\in\{B,C,K\}\) and \(d_R\) (resp. \(d_B\), \(d_C\), \(d_K\)) denotes the distance in the \(\Lambda\)- (resp. Bergman, Carathéodory, Kobayashi) metric in \(D\). NEWLINENEWLINENEWLINE 4. Let \(D\) be \(\mathcal C^2\)-smooth strongly pseudoconvex, \(z_0\in\partial D\), \(v\in\mathbb C^n\). ThenNEWLINE NEWLINE\[NEWLINE \lim_{z\to z_0}R\big(z,v_N(z)\big)=\frac{1}{n-1}, NEWLINE\]NEWLINE NEWLINEwhere \(R\) denotes the holomorphic sectional curvature. NEWLINENEWLINENEWLINE 5. Let \(D\) be \(\mathcal C^{\infty}\)-smooth strongly convex. Suppose that the \(\Lambda\)-metric in \(D\) has constant negative holomorphic sectional curvature. Then \(D\) is biholomorphic to the unit ball and the \(\Lambda\)-metric is proportional to the Bergman metric.