Notes on Short $\mathbb{C}^k$'s

TL;DR

This paper investigates special complex domains called Short ^ks, showing they have infinite volume, trivial Bergman spaces, and are typically Loewner, with new examples constructed via non-autonomous basins of attraction.

Contribution

It introduces Loewner Short ^ks, explores their properties, and constructs new examples, advancing understanding of these complex domains and their biholomorphic classifications.

Findings

Short ^ks have infinite volume and trivial Bergman spaces.

Most Short ^ks are Loewner, meaning they can be exhausted by biholomorphic images of the unit ball.

Constructed new examples of Short ^ks using non-autonomous basins of attraction.

Abstract

Domains that are increasing union of balls (up to biholomorphism) and on which the Kobayashi metric vanishes identically arise inexorably in complex analysis. In this article we show that in higher dimensions these domains have infinite volume and the Bergman spaces of these domains are trivial. As a consequence they fail to be strictly pseudo-convex at each of their boundary points although these domains are pseudo-convex by definition. These domains can be of different types and one of them is Short $\mathbb{C}^k$'s. In pursuit of identifying the Runge Short $\mathbb{C}^k$'s (up to biholomorphism), we introduce a special class of Short $\mathbb{C}^k$'s, called Loewner Short $\mathbb{C}^k$'s. These are those Short $\mathbb{C}^k$'s which can be exhausted in a continuous manner by a strictly increasing parametrized family of open sets, each of which is biholomrphically equivalent to the unit ball and therefore, they are Runge up to biholomorphism. Although, the question of whether all Short $\mathbb{C}^k$'s are Runge (up to biholomorphism), or whether all Short $\mathbb{C}^k$'s are Loewner remains unsettled, we show that the typical Short $\mathbb{C}^k$'s are Loewner. In the final section, we construct a bunch of non-autonomous basins of attraction, which serve as interesting examples of Short $\mathbb{C}^2$'s.