Increasing sequences of complex manifolds with uniform squeezing constants and their Bergman spaces

TL;DR

This paper characterizes complex manifolds formed by increasing unions of bounded domains with uniform squeezing constants, linking their structure to the Kobayashi metric and determining the dimension of their Bergman spaces.

Contribution

It provides a classification of such manifolds based on Kobayashi corank and proves that their Bergman space dimension is either zero or infinite, confirming Wiegerinck's conjecture in this setting.

Findings

Manifolds with full Kobayashi corank are unions of unit balls.

Manifolds with zero Kobayashi corank admit bounded realizations with uniform squeezing.

Intermediate corank manifolds have a local weak vector bundle structure.

Abstract

For $d\geq 2$, we discuss $d$-dimensional complex manifolds $M$ that are the increasing union of bounded open sets $M_n$'s of $\mathbb{C}^d$ with a common uniform squeezing constant. The description of $M$ is given in terms of the corank of the infinitesimal Kobayashi metric of $M$, which is shown to be identically constant on $M$. The main result of this article says that if $M$ has full Kobayashi corank, then $M$ can be written as an increasing union of the unit ball; if $M$ has zero Kobayashi corank, then $M$ has a bounded realization with a uniform squeezing constant; and if $M$ has an intermediate Kobayashi corank, then $M$ has a local weak vector bundle structure. The above description of $M$ is used to show that the dimension of the Bergman space of $M \subseteq \mathbb{C}^d$ is either zero or infinity. This settles Wiegerinck's conjecture for those pseudoconvex domains in higher dimensions that are increasing union of bounded domains with a common uniform squeezing constant.