  It is shown that any open Riemann surface can be immersed in any Stein
manifold with (volume) density property and of dimension at least 2, if the
manifold possesses an exhaustion with holomorphically convex compacts such that
their complement is connected. The immersion can be made into an embedding if
the dimension is at least 3. As an application, it is shown that Stein
manifolds with (volume) density property and of dimension at least 3, are
characterized among all other complex manifolds by their semigroup of
holomorphic endomorphisms.
