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

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 Carathéodory and the Kobayashi metrics on these domains.