Variations of Bergman Kernels for Some Explicitly Given Families of Planar Domains

TL;DR

This paper investigates how Bergman kernels vary with parameters across different families of planar domains, revealing diverse boundary behaviors depending on domain smoothness and geometry.

Contribution

It provides explicit examples showing how the parameter dependence of Bergman kernels differs between smooth and non-smooth domains, highlighting new phenomena.

Findings

Levi form approaches 0 for annuli as boundary is approached

Levi form approaches 1 for discs as boundary is approached

Non-smooth boundaries exhibit distinct, different behaviors

Abstract

We study the parameter dependence of the Bergman kernels on some planar domains depending on complex parameter \zeta in nontrivial "pseudoconvex" ways. Smoothly bounded cases are studied at first: It turns out that, in an example where the domains are annuli, the Levi form for the logarithm of the Bergman kernels with respect to \zeta approaches to 0 as the point tends to the boundary of the domain, and in another example where the domains are discs, it approaches to 1 as the point tends to the complement of a point in the boundary. Further, in contrast to this, in the cases where the boundary of the domains are not smooth, such as discs with slits, rectangles and half strips, completely different phenomena are observed.