Mergelyan's and Arakelian's theorems for manifold-valued maps

TL;DR

This paper extends classical approximation theorems to manifold-valued maps, demonstrating that Mergelyan's theorem applies to open Riemann surfaces and Oka manifolds, and proving an Arakelian-type approximation result for complex homogeneous manifolds.

Contribution

It establishes the validity of Mergelyan's theorem for manifold-valued maps and derives an Arakelian-type approximation theorem for maps into complex homogeneous manifolds.

Findings

Mergelyan's theorem holds for maps from open Riemann surfaces to Oka manifolds.

An analogue of Arakelian's theorem is proven for uniform approximation of holomorphic maps into compact complex homogeneous manifolds.

The results extend classical approximation theorems to a broader class of target manifolds.

Abstract

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 the plane $\mathbb C$ to any compact complex homogeneous manifold.