2021/04/26 by John Erik Fornæss, Fornaess, John Erik, Ratna Pal +1 · 1 citation
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2104.12413
openalex publication_date 2021/04/26 · openalex created_date 2021/05/10 · openalex updated_date 2026/07/28
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 ℂk's. In pursuit of identifying the Runge Short ℂk's (up to biholomorphism), we introduce a special class of Short ℂk's, called Loewner Short ℂk's. These are those Short ℂk'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 ℂk's are Runge (up to biholomorphism), or whether all Short ℂk's are Loewner remains unsettled, we show that the typical Short ℂk's are Loewner. In the final section, we construct a bunch of non-autonomous basins of attraction, which serve as interesting examples of Short ℂ2's.