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

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 (sic) C-n and any connected complex manifold Y, the space 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 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 (sic) C-n, any fixed-point-free automorphism of Omega and any connected complex manifold Y, there exists a universal map Omega -> Y. We also characterize Oka manifolds by the existence of universal maps.