Embedding univalent functions in filtering Loewner chains in higher dimension

TL;DR

This paper investigates conditions under which univalent functions in higher dimensions can be embedded into Loewner chains, establishing a link between polynomial convexity of the image and embeddability.

Contribution

It provides a necessary and sufficient condition for embedding univalent maps into Loewner chains in higher dimensions based on polynomial convexity.

Findings

Embedding is possible if and only if the image's closure is polynomially convex.

The result applies to normalized univalent maps of the ball with smooth strongly pseudoconvex images.

The paper extends Loewner theory to higher dimensions with regularity conditions.

Abstract

We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is embeddable into a normalized Loewner chain (satisfying also some extra regularity properties) if and only if the closure of the image is polynomially convex.