Complete meromorphic curves with Jordan boundaries

Abstract

We prove that given a finite set $E$ in a bordered Riemann surface $\mathcal{R}$, there is a continuous map $h:\overline{\mathcal{R}}\setminus E\to {\mathbb{C}}^n$ ($n\ge 2$) such that $h{|}_{\mathcal{R}\setminus E}:\mathcal{R}\setminus E\to {\mathbb{C}}^n$ is a complete holomorphic immersion (embedding if $n\ge 3$) which is meromorphic on $\mathcal{R}$ and has effective poles at all points in $E$, and $h{|}_{b\overline{\mathcal{R}}}:b\overline{\mathcal{R}}\to {\mathbb{C}}^n$ is a topological embedding. In particular, $h(b\overline{\mathcal{R}})$ consists of the union of finitely many pairwise disjoint Jordan curves which we ensure to be of Hausdorff dimension one. We establish a more general result including uniform approximation and interpolation.

To appear in Journal of Mathematical Analysis and Applications