Lower estimates of the Kobayashi distance and limits of complex geodesics

TL;DR

This paper establishes lower bounds for the Kobayashi distance in strongly pseudoconvex domains with smooth boundaries, analyzing the behavior of complex geodesics near the boundary and their limits.

Contribution

It provides new estimates for the Kobayashi distance by studying complex geodesics approaching the boundary in strongly pseudoconvex domains.

Findings

Lower bounds for the Kobayashi distance are derived.

Complex geodesics with non-tangential differences intersect a fixed compact set.

Results depend only on the rate of non-tangentiality.

Abstract

It is proved for a strongly pseudoconvex domain $D$ in $\Bbb C^d$ with $\mathcal C^{2,\alpha}$-smooth boundary that any complex geodesic through every two close points of $D$ sufficiently close to $\partial D$ and whose difference is non-tangential to $\partial D$ intersect a compact subset of $D$ that depends only on the rate of non-tangentiality. As an application, a lower bound for the Kobayashi distance is obtained.