# Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps

## TL;DR

This paper investigates the existence of dense holomorphic curves in spaces of holomorphic maps, characterizes Oka manifolds via universal maps, and applies these results to the theory of universal functions in complex analysis.

## Contribution

It establishes conditions for dense holomorphic curves in mapping spaces and characterizes Oka manifolds through the existence of universal maps, advancing the understanding of complex geometric structures.

## Key findings

- Existence of dense holomorphic discs in $ ext{O}( ext{Omega},Y)$ for convex domains and connected manifolds.
- Characterization of Oka manifolds via dense entire curves in mapping spaces.
- Construction of universal maps from convex domains to complex manifolds.

## Abstract

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain $\Omega\Subset\mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc. Our second result states that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$. In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain $\Omega\Subset\mathbb{C}^n$, any fixed-point-free automorphism of $\Omega$ and any connected complex manifold $Y$, there exists a universal map $\Omega\to Y$. We also characterize Oka manifolds by the existence of universal maps.