Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps
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 fixedpointfree 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2017
 DOI:
 10.48550/arXiv.1702.08022
 arXiv:
 arXiv:1702.08022
 Bibcode:
 2017arXiv170208022K
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Dynamical Systems;
 32E10;
 32H02;
 54C35;
 30K20;
 47A16
 EPrint:
 15 pages