Analysis and geometry on worm domains

TL;DR

This paper explores the properties of the Bergman kernel on worm domains, providing asymptotic expansions, boundary irregularities, and mapping properties, with extensive background and discussion of smooth variants.

Contribution

It offers a comprehensive analysis of the Bergman kernel on worm domains, including new asymptotic expansions and boundary irregularity results.

Findings

Asymptotic expansion of the Bergman kernel

Boundary irregularity properties established

Mapping properties of the Bergman projection analyzed

Abstract

We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included.