Remarks on the metric induced by the Robin function

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using $G(z, p)$, the Green function for $D$ with pole at $p \in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a K\"{a}hler metric (the so-called $\La$-metric) using the Robin function $\La(p)$ arising from $G(z, p)$. The purpose of this article is to study this metric by deriving its boundary asymptotics and using them to calculate the holomorphic sectional curvature along normal directions. It is also shown that the $\La$-metric is comparable to the Kobayashi (and hence to the Bergman and Carath\'{e}odory metrics) when $D$ is strongly pseudoconvex. The unit ball in $\mathbf C^n$ is also characterized among all smoothly bounded strongly convex domains on which the $\La$-metric has constant negative holomorphic sectional curvature. This may be regarded as a version of Lu-Qi Keng's theorem for the Bergman metric.