An embedding of the unit ball that does not embed into a Loewner chain

TL;DR

The paper constructs a specific holomorphic embedding of the 3-dimensional unit ball into complex 3-space that cannot be extended into a larger domain as a Runge domain, impacting the understanding of Loewner chains.

Contribution

It provides a counterexample showing the embedding of the unit ball in three dimensions does not embed into any larger domain as a Runge domain, revealing new limitations in complex analysis.

Findings

Constructed a holomorphic embedding of -dimensional unit ball

Demonstrated the embedding is not Runge in any larger domain

Showed that -dimensional case differs from lower dimensions

Abstract

We construct a holomorphic embedding $\phi:\mathbb B^3\rightarrow\mathbb C^3$ such that $\phi(\mathbb B^3)$ is not Runge in any strictly larger domain. As a consequence, $\mathcal S\neq\mathcal S^1$ for $n=3$.