Remarks on the metric induced by the Robin function

TL;DR

This paper investigates the properties of the -metric derived from the Robin function on smoothly bounded pseudoconvex domains, analyzing its boundary behavior, curvature, and relation to classical metrics like Kobayashi and Bergman.

Contribution

It provides boundary asymptotics for the -metric, compares it to other metrics on strongly pseudoconvex domains, and characterizes the unit ball via constant negative curvature.

Findings

-metric is comparable to Kobayashi, Bergman, and Carathe9odory metrics on strongly pseudoconvex domains.

Boundary asymptotics of the -metric are derived.

The unit ball is characterized among strongly convex domains by constant negative holomorphic sectional curvature.

Abstract

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.