Embedding subsets of tori Properly into $\CC^2$

TL;DR

This paper proves that any subset of a torus with finitely many boundary components can be properly embedded into complex two-dimensional space, expanding understanding of embeddings of complex manifolds.

Contribution

It establishes the proper embedding of subsets of tori with finitely many boundary components into a22, a result not previously known for such topological configurations.

Findings

All such subsets embed properly into a22.

The result applies to subsets with boundary components that are not points.

This advances the theory of complex embeddings of topological surfaces.

Abstract

Let $\TT$ be a torus. We prove that all subsets of $\TT$ with finitely many boundary components (none of them being points) embed properly into $\CC^2$.