Complete meromorphic curves with Jordan boundaries

Geometry seminar

Speaker: Tjaša Vrhovnik

Abstract:We prove that given a finite set \(E\) in a bordered Riemann surface $\mathcal{R}$, there is a continuous map \(h\colon \overline{\mathcal{R}}\setminus E\to\mathbb{C}^n\) (\(n\geq 2\)) such that \(h|_{\mathcal{R}\setminus E} \colon \mathcal{R}\setminus E\to\mathbb{C}^n\) is a complete holomorphic immersion (embedding if \(n\geq 3\)) which is meromorphic on \(\mathcal{R}\) and has effective poles at all points in \(E\), and \(h|_{b\overline{\mathcal{R}}} \colon b\overline{\mathcal{R}}\to\mathbb{C}^n\) 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.