The embedding conjecture and the approximation conjecture in higher dimension

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.