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

John Erik Fornæss, Ratna Pal

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.

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

Abstract

For , we discuss -dimensional complex manifolds that are the increasing union of bounded open sets 's of with a common uniform squeezing constant. The description of is given in terms of the corank of the infinitesimal Kobayashi metric of , which is shown to be identically constant on . The main result of this article says that if has full Kobayashi corank, then can be written as an increasing union of the unit ball; if has zero Kobayashi corank, then has a bounded realization with a uniform squeezing constant; and if has an intermediate Kobayashi corank, then has a local weak vector bundle structure. The above description of is used to show that the dimension of the Bergman space of 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.
Paper Structure (7 sections, 7 theorems, 37 equations)

This paper contains 7 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. $M$ with full Kobayashi corank at some point
  3. Proof of [(I), Theorem \ref{['main thm']}]
  4. $M$ with zero Kobayashi corank at some point
  5. Proof of [(II), Theorem \ref{['main thm']}]
  6. $M$ with intermediate Kobayashi corank at some point
  7. Proof of [(III), Theorem \ref{['main thm']}]

Key Result

Theorem 1.5

For $d\geq 2$ and for each $n\geq 1$, let $M_n$ be biholomorphic to (possibly different) bounded open sets in $\mathbb{C}^d$ such that $M_n\subset \subset M_{n+1}$ and suppose $M=\bigcup_{n=1}^\infty M_n$. Suppose the family $\{M_n\}_{n\geq1}$ has a uniform squeezing constant, say $r>0$. Then for ea

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 14 more