In this paper we show that Mergelyan's theorem holds for maps from open Riemann surfaces to Oka manifolds. This is used to prove the analogue of Arakelian's theorem on uniform approximation of holomorphic maps from closed subsets of plane domains to any compact complex homogeneous manifold.
COBISS.SI-ID: 18698073
Let $D \Subset \mathbb{C}^N$ be a domain with smooth boundary, of finite 1-type at a point $p \in bD$ and such that the $\overline{D}$ has a basis of Stein Runge neighborhoods. Assume that there exists an analytic disc which intersects $\overline{D}$ exactly at $p$. We construct proper holomorphic maps from any open Riemann surface $S$ to $\mathbb{C}^N$ which are attached to the $\overline{D}$ exactly at $p$.
COBISS.SI-ID: 18647129
We provide a direct construction of Poletsky discs via local arc approximation and a Runge-type theorem by A. Gournay [Geom Funct. Anal. 22 (2012), pp. 311-351].
COBISS.SI-ID: 18478169