Existence of geodesic spirals for the Kobayashi--Fuks metric on planar domains

TL;DR

This paper investigates the existence of geodesic spirals in the interior of certain complex domains under the Kobayashi--Fuks metric, providing an affirmative answer for non-simply connected planar domains.

Contribution

It establishes the existence of geodesic spirals in non-simply connected planar domains equipped with the Kobayashi--Fuks metric, a problem previously unresolved.

Findings

Existence of geodesic spirals in non-simply connected planar domains.

Positive answer for the case when the domain is not simply connected.

Focus on the Kobayashi--Fuks metric in complex analysis.

Abstract

In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain $D$ in $\mathbb{C}^n$, can we guarantee the existence of geodesics for the Kobayashi--Fuks metric which ``spiral around" in the interior of $D$? We find an affirmative answer to the above question for $n=1$ when $D$ is not simply connected.