Approximation and accumulation results of holomorphic mappings with dense image

TL;DR

This paper presents four approximation theorems for manifold-valued holomorphic mappings, demonstrating how to approximate embeddings and mappings with dense images in various complex settings.

Contribution

It introduces new approximation theorems for holomorphic maps with dense images, expanding understanding of holomorphic embeddings and mappings in complex manifolds.

Findings

Approximation of holomorphic embeddings with dense images on pseudoconvex domains.

Approximation of bounded holomorphic mappings by those with dense images.

Construction of sequences of holomorphic mappings converging to non-holomorphic dense image mappings.

Abstract

We display four approximation theorems for manifold-valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\Bbb C^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).