Complete meromorphic curves with Jordan boundaries

We prove that given a finite set Ein a bordered Riemann surface R, there is a continuous map h: (R) over barE -> C-n (n >= 2) such that h vertical bar(R\E): R \ E -> C-n 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 vertical bar(b (R) over bar): b (R) over bar -> C-n is a topological embedding. In particular, h(b (R) over bar) 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. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

We prove the existence of a continuous map on a bordered Riemann surface which is holomorphic at some points, has effective poles at other points, and is a topological embedding on the boundary.