We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. We also construct a Stein neighborhood basis of any compact complex curve with C 2 boundary in a complex space.

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in R n and null holomorphic curves in C n for any n ≥ 3. With this tool in hand we construct complete conformally immersed minimal surfaces in R n which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of R n which extend continuously up to the boundary; for n ≥ 5 we find embeddings with these properties.

In this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold $X$ is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on $X$. Examples show that our conditions cannot be weakened in general. We obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with Griffiths positive normal bundle; in the projective case these generalize classical theorems of Remmert, Bishop and Narasimhan concerning proper holomorphic maps and embeddings to ${\Bbb C}^n ={\Bbb P}^n \backslash {\Bbb P}^{n-1}$..