An approximate Riemann mapping theorem in \({\mathbb{C}}^ n\) (Q1067053)

Let \(T_ 0(n)\) be the class of all domains in \({\mathbb{C}}^ n\) diffeomorphic to the unit ball. The following theorem is proved. For any two domains \(G_ i\in T_ 0(n)\) and compacts \(K_ i\subset G_ i\), \(i=1,2\) there exist a domain \(D\in T_ 0(n)\) and two biholomorphic imbeddings \(F_ i: D\to G_ i\) such that \(F_ i(D)\supset K_ i\), \(i=1,2.\) The domain D in this theorem is constructed independently of \(G_ i\) and \(K_ i.\) The paper contains also an example of a diffeomorphism class for which the statement of the theorem does not hold.