Complete meromorphic curves with Jordan boundaries

Abstract

We prove that given a finite set E in a bordered Riemann surface R, there is a continuous map h:R‾∖E→Cn (n≥2) such that h|R∖E:R∖E→Cn is a complete holomorphic immersion (embedding if n≥3) which is meromorphic on R and has effective poles at all points in E, and h|bR‾:bR‾→Cn is a topological embedding. In particular, h(bR‾) 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.