An example for the holomorphic sectional curvature of the Bergman metric

TL;DR

This paper investigates the behavior of the holomorphic sectional curvature of the Bergman metric on planar annuli, revealing divergence at boundary points and extremal curvature limits.

Contribution

It constructs a domain where the curvature diverges at a boundary point and demonstrates the extremal limits of the Bergman metric's curvature behavior.

Findings

Curvature diverges at boundary points of certain domains.

Limes superior of curvature reaches the maximum value 2.

Limes inferior of curvature is -infinity.

Abstract

In this paper we study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreover the limes superior is the maximal possible for the Bergman metric (2), whereas the limes inferior is $-\infty$.