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Non-Runge Fatou-Bieberbach Domains in Stein Manifolds with the Density Property

Gaofeng Huang, Frank Kutzschebauch, Feng Rong

Abstract

Let $X$ be a Stein manifold with the density property. We present methods of constructing two kinds of non-Runge Fatou-Bieberbach domains in $X$, which by definition are proper open subsets of $X$ biholomorphic either to $\mathbb{C}^n$ or to $X$. For both kinds, we provide examples where our methods apply.

Non-Runge Fatou-Bieberbach Domains in Stein Manifolds with the Density Property

Abstract

Let be a Stein manifold with the density property. We present methods of constructing two kinds of non-Runge Fatou-Bieberbach domains in , which by definition are proper open subsets of biholomorphic either to or to . For both kinds, we provide examples where our methods apply.
Paper Structure (9 sections, 13 theorems, 40 equations, 1 figure)

This paper contains 9 sections, 13 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak density property
  4. Fatou-Bieberbach domains of the first kind
  5. Fatou-Bieberbach domains of the second kind
  6. Examples
  7. The first kind
  8. The second kind
  9. A topological observation

Key Result

Theorem 2.2

Let $X$ be a Stein manifold with the density property. If $\Phi_t : \Omega \stackrel{\cong}{\to} \Omega_t=\Phi_t(\Omega)\subset X$$(t\in [0,1])$ is a continuous isotopy of biholomorphic maps between Stein Runge domains in $X$ where $\Phi_0: \Omega \to X$ is the inclusion map, then $\Phi_1$ can be ap

Figures (1)

  • Figure 1: Illustration of Property (PO)

Theorems & Definitions (29)

  • Definition 2.1: Varolin MR1829353
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4: Kutzschebauch-Leuenberger-Liendo MR3320241
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • Remark 2.8
  • ...and 19 more