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

Yuta Kusakabe

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

E-mail Address: y-kusakabe@cr.math.sci.osaka-u.ac.jp

https://doi.org/10.1142/S0129167X17500288Cited by:5 (Source: Crossref)

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 $Ω ⋐ ℂ^{n}$ and any connected complex manifold $Y$ , the space $𝒪 (Ω, Y)$ 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 $𝒪 (X, Y)$ .

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 $Ω ⋐ ℂ^{n}$ , any fixed-point-free automorphism of $Ω$ and any connected complex manifold $Y$ , there exists a universal map $Ω \to Y$ . We also characterize Oka manifolds by the existence of universal maps.

Keywords:

AMSC: 32E10, 32H02, 54C35, 30K20, 47A16

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