Remarks on the metric induced by the Robin function III

TL;DR

This paper investigates the existence of geodesic spirals in the mbda-metric, a Ke4hler metric derived from the Robin function associated with the Green function on smoothly bounded pseudoconvex domains in complex space.

Contribution

It extends previous work by analyzing geodesic spirals in the mbda-metric, providing new insights into its geometric properties.

Findings

Existence of geodesic spirals in the mbda-metric established.

Conditions under which geodesic spirals occur are identified.

The study enhances understanding of the metric's geometric structure.

Abstract

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using the Robin function $\Lambda(p)$ that arises from the Green function $G(z, p)$ 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 $\Lambda$-metric) on $D$. In this article, we study the existence of geodesic spirals for this metric.