Complete embedded complex curves in the ball of $\mathbb{C}^2$ can have any topology

TL;DR

This paper proves that the unit ball in complex two-space can contain complete, properly embedded complex curves of any topology, including those with any specified closed discrete subset, expanding understanding of complex curve embeddings.

Contribution

It demonstrates the existence of complete properly embedded complex curves of arbitrary topology within the unit ball of ^2, including those with prescribed discrete subsets.

Findings

Existence of complete properly embedded complex curves of any topology in ^2

Construction of curves containing any given closed discrete subset

Advancement in understanding of complex curve embeddings in complex balls

Abstract

In this paper we prove that the unit ball $\mathbb{B}$ of $\mathbb{C}^2$ admits complete properly embedded complex curves of any given topological type. Moreover, we provide examples containing any given closed discrete subset of $\mathbb{B}$.