  In this paper (a sequel to B. Drinovec Drnovsek and F. Forstneric,
Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203-253) 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 is expressed in
terms of the Morse indices and the number of positive Levi eigenvalues of an
exhaustion function on the manifold. Examples show that our condition cannot be
weakened in general. Optimal results are obtained for subvarieties of this type
in complements of compact complex submanifolds with Griffiths positive normal
bundle; in the projective case these results generalize classical theorems of
Remmert, Bishop and Narasimhan concerning proper holomorphic maps and
embeddings to complex Euclidean spaces.
