The embedding conjecture and the approximation conjecture in higher dimension

TL;DR

This paper establishes the equivalence of three major conjectures in higher-dimensional complex analysis, linking embedding, approximation, and immersion problems related to univalent maps and Fatou-Bieberbach domains.

Contribution

It proves the equivalence among the embedding conjecture, the approximation conjecture, and the immersion conjecture in higher dimensions.

Findings

Proves the equivalence of three fundamental conjectures in complex analysis.

Connects univalent map embedding, approximation, and domain immersion problems.

Provides a unified framework for understanding these conjectures in higher-dimensional spaces.

Abstract

In this paper we show the equivalence among three conjectures (and related open questions), namely, the embedding of univalent maps of the unit ball into Loewner chains, the approximation of univalent maps with entire univalent maps and the immersion of domain biholomorphic to the ball in a Runge way into Fatou-Bieberbach domains.