Asymptotic behaviour of the sectional curvature of the Bergman metric for annuli

TL;DR

This paper analyzes the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli, simplifying previous results and constructing a domain with extreme curvature properties.

Contribution

It extends and simplifies prior work on Bergman metric curvature in annuli and constructs a domain with unbounded curvature extremities.

Findings

Supremum of curvature is two in the constructed domain.

Infimum of curvature is negative infinity.

Provides a simplified approach to previous asymptotic results.

Abstract

We extend and simplify results of \cite{Din~2009} where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly as in \cite{Din~2009} the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two whereas its infimum is equal to $-\infty$.