Asymptotic behavior of the Kobayashi metric in the normal direction

TL;DR

This paper constructs a specific pseudoconvex domain in complex three-dimensional space demonstrating that the Kobayashi metric's growth rate in the normal direction can deviate from the typical inverse distance behavior.

Contribution

It provides a counterexample showing the Kobayashi metric does not always blow up at the expected rate in the normal direction in pseudoconvex domains.

Findings

Kobayashi metric can have non-standard asymptotic behavior

Counterexample in $\\mathbb C^3$ for normal boundary behavior

Challenges assumptions about metric growth in complex analysis

Abstract

In this paper, we construct a pseudoconvex domain in $\mathbb C^3$ where the Kobayashi metric does not blow up at a rate of one over distance to the boundary in the normal direction.