Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains

TL;DR

This paper investigates the relationship between invariant metrics on strongly pseudoconvex and worm domains, showing their equivalence near boundary points and exploring complex geodesics and boundary properties.

Contribution

It demonstrates the coincidence of Carathéodory and Kobayashi metrics near boundary points and extends results to worm domains, including boundary exposure and geodesic uniqueness.

Findings

Metrics coincide near boundary points in strongly pseudoconvex domains.

Existence and uniqueness of complex geodesics near boundary points.

Strongly pseudoconvex boundary points of worm domains can be globally exposed.

Abstract

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carath\'eodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is sufficiently close to being tangential to $bD$. Also, we show that every two close points of $D$ sufficiently close to the boundary and whose difference is almost tangential to $bD$ can be joined by a (unique up to reparameterization) complex geodesic of $D$ which is also a holomorphic retract of $D$. The same continues to hold if $D$ is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed, this has consequences for the behavior of the squeezing function.