This is joint work with Barbara Drinovec-Drnovsek.
Let X be an irreducible Stein space of dimension >1 or,
more generally, an irreducible q-convex complex space
with q < dim X. We prove that the interior of any
finite bordered Riemann surface D can be represented
as a closed complex curve in X. This is not true in
general without the q-convexity assumption. Our
construction combines a new Cartan type splitting lemma
with estimates up to the boundary and a local solution
of a Riemann-Hilbert boundary value problem.