## An approximate Riemann mapping theorem in $${\mathbb{C}}^ n$$.(English)Zbl 0579.32041

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.

### MSC:

 32H99 Holomorphic mappings and correspondences 30C25 Covering theorems in conformal mapping theory 32A30 Other generalizations of function theory of one complex variable 14E25 Embeddings in algebraic geometry
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### References:

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