Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls

TL;DR

This paper constructs a complete proper holomorphic embedding of any strictly pseudoconvex domain with smooth boundary into a high-dimensional unit ball, solving a previously open problem in complex analysis.

Contribution

It provides a method to embed strictly pseudoconvex domains into balls, extending the understanding of complex structures and embedding possibilities.

Findings

Successful construction of embeddings for all such domains

Answer to a longstanding open question

Extension of embedding techniques to higher dimensions

Abstract

We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.