On the geodesics of the Szeg\"o metric

TL;DR

This paper investigates the properties of geodesics, including closed geodesics and spirals, in the Szeg"o metric on complex domains that are smooth, strongly pseudoconvex, and not simply connected, extending understanding in complex geometry.

Contribution

It provides new insights into the existence and behavior of geodesics in the Szeg"o metric on complex domains with specific topological and geometric properties.

Findings

Existence of closed geodesics in the Szeg"o metric.

Presence of geodesic spirals in non-simply connected domains.

Extension of geodesic theory to complex domains with strong pseudoconvexity.

Abstract

We explore the existence of closed geodesics and geodesic spirals for the Szeg\"o metric in a $C^{\infty}$-smoothly bounded strongly pseudoconvex domain $\Omega\subset\mathbb{C}^n$, which is not simply connected for $n \geq 2$.