Boundary behavior of the Kobayashi distance in pseudoconvex Reinhardt domains

TL;DR

This paper investigates the asymptotic behavior of the Kobayashi distance near the boundary of pseudoconvex Reinhardt domains, providing upper and lower bounds under various smoothness and convergence conditions.

Contribution

It establishes new boundary growth estimates for the Kobayashi distance in pseudoconvex Reinhardt domains, including sharp bounds under smoothness assumptions.

Findings

Kobayashi distance grows at most like -log d_D+C near boundary

Growth does not exceed 1/2 log(-log d_D)+C for certain boundary points

Lower bounds are established under smoothness and non-tangential convergence conditions

Abstract

We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain $D$ increases asymptotically at most like $-\log d_D+C$. Moreover, for boundary points from $\text{int}\bar{D}$ the growth does not exceed $1/2\log(-\log d_D)+C$. The lower estimate by $-1/2\log d_D+C$ is obtained under additional assumptions of $\mathcal C^1$-smoothness of a domain and a non-tangential convergence.