Quantitative localization and comparison of invariant distances of domains in $\mathbb C^n$

TL;DR

This paper provides explicit bounds and sharp estimates for comparing local and global invariant distances, specifically Kobayashi distances, in complex domains, enhancing understanding of their boundary behavior in several complex variables.

Contribution

It introduces explicit bounds on the difference and ratio between local and global Kobayashi distances in complex domains, with sharp estimates in one dimension.

Findings

Explicit bounds on Kobayashi distance differences and ratios.

Sharp estimates for invariant distances in one complex dimension.

Enhanced understanding of boundary behavior of invariant metrics.

Abstract

We obtain explicit bounds on the difference and ratio between "local" and "global" Kobayashi distances in a domain of $\mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension $1$.