Boundary behaviour of the Span metric and its higher-order curvatures

TL;DR

This paper investigates the boundary behavior of the span metric and its higher-order curvatures on finitely connected Jordan domains, providing sharp estimates and comparisons with other classical metrics.

Contribution

It introduces a scaling-based approach to analyze the boundary behavior of the span metric and establishes its comparability to Carathéodory and Kobayashi metrics on smooth domains.

Findings

Boundary behavior characterized using scaling principles

Localization of the span metric near boundary points

Establishment of metric comparability on smooth domains

Abstract

In this note, we use scaling principle to study the boundary behaviour of the span metric and its higher-order curvatures on finitely connected Jordan planar domains. A localization of this metric near boundary points of finitely connected Jordan domains is also obtained. Further, we obtain boundary sharp estimates for this metric on $ C^2 $-smooth bounded domains and consequently, this metric is comparable to the Carath\'eodory and the Kobayashi metrics on these domains.