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

TL;DR

This paper investigates certain invariant domains in complex n-space, proving they are Runge, and shows biholomorphisms can be approximated by automorphisms, with applications to Loewner PDE solutions and volume-preserving maps.

Contribution

It establishes new approximation theorems for biholomorphisms on invariant domains, generalizing previous results and applying to Loewner PDEs and volume-preserving maps.

Findings

Domains are always Runge.

Biholomorphisms can be approximated by automorphisms.

Existence of unique solutions to Loewner PDEs in these domains.

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 $\Omega$ is always Runge. Next, with an additional assumption on the rate of convergence of the flow, we show that any biholomorphism $\Phi\colon \Omega \to \Phi(\Omega)$, with $\Phi(\Omega)$ 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 $\Omega$ 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.