Rational approximation on ares in
We show that any open Riemann surface can be properly immersed in any Stein manifold with the (Volume) Density property and of dimension at least 2. If the dimension is at least 3, we can actually choose this immersion to be an embedding. As an application, we show that Stein manifolds with the (Volume) Density property and of dimension at least 3, are characterized among all other complex manifolds by their semigroup of holomorphic endomorphisms.
The general Stein union problem is solved: given an increasing sequence of Stein open sets, it is shown that the union is Stein if and only if is Hausdorff separated.