Approximations on certain domains of $\mathbb{C}^{n}$

TL;DR

This paper develops a framework to study domains in $\mathbb{C}^n$ that are invariant under the positive-time flows of complete holomorphic vector fields with globally attracting fixed points. Under explicit exponential decay conditions on the flow, the authors prove that such domains are Runge and that any biholomorphism with Runge image can be uniformly approximated by automorphisms of $\mathbb{C}^n$, extending prior results beyond linear dynamics. They apply these approximation results to Loewner PDE on complete hyperbolic domains, obtaining essentially unique univalent solutions in $\mathbb{C}^n$ and establishing Runge/Stein targets for the evolution, along with volume-preserving and symplectic variants. The work unifies and generalizes several strands of Andersén–Lempert theory, formalizes growth conditions via spectral data of $DV(0)$, and provides diverse examples illustrating spirallike domains and their approximation properties, thereby broadening the toolkit for several complex variables and Loewner theory.

Abstract

In this paper, we study the domains in $\mathbb{C}^n$ that are invariant under the positive flows of some globally defined, complete holomorphic vector field with a globally attracting fixed point at the origin. Our first result says that such a domain $Ω$ is always Runge. Next, with an additional assumption on the rate of convergence of the flow, we show that any biholomorphism $Φ\colon Ω\to Φ(Ω)$, with $Φ(Ω)$ is Runge, can be approximated by automorphisms of $\mathbb{C}^{n}$ uniformly on compacts. This generalizes all earlier known theorems in this direction substantially, even when the vector field is linear. As an application of our approximation results, on such domains that are also complete hyperbolic, we show that any Loewner PDE in a complete hyperbolic domain $Ω$ admits an essentially unique univalent solution with values in $\mathbb{C}^n$. We also provide an approximation result for volume preserving biholomorphisms on above domains. We provide several examples of such domains.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.5
• Remark 1.6
• Corollary 1.7
• Corollary 1.8
• Conjecture 1.9
• Theorem 1.10
• Theorem 1.11
• Remark 2.1
• Lemma 2.7