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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:

[1] Alexander, H.: Extremal holomorphic imbeddings between ball and polydisc. Proc. Am. Math. Soc.68, 200-202 (1978) · Zbl 0379.32022
[2] Bedford, E.: Proper holomorphic mappings. Bull. Am. Math. Soc.10, 157-175 (1984) · Zbl 0534.32009
[3] Burns, D., Shnider, S., Wells, R.: On deformations of strictly pseudoconvex domains. Invent. Math.46, 237-253 (1978) · Zbl 0412.32022
[4] Fornaess, J.-E., Stout, E.L.: Polydiscs in complex manifolds. Math. Ann.227, 145-153 (1977) · Zbl 0338.32008
[5] Fridman, B.L.: A universal exhausting domain. Proc. Am. Math. Soc. (to appear) · Zbl 0605.32012
[6] Goluzin, G.M.: Geometric theory of functions of a complex variable, GITTL, Moscow, 1952; English transl., Transl. Math. Monographs, Vol. 26. Providence: Amer. Math. Soc. 1969 · Zbl 0049.05902
[7] Greene, R., Krantz, S.: Deformation of complex structures, estimates for the \(\bar \partial\) equation and stability of the Bergman Kernel. Adv. Math.43, 1-86 (1982) · Zbl 0504.32016
[8] Krantz, S.G.: Function theory of several complex variables. New York: Wiley 1982 · Zbl 0471.32008
[9] Lempert, L.: A note on mapping polydiscs into balls and vice versa. Acta. Math. Hung.34, 117-119 (1979) · Zbl 0424.32001
[10] Rudin, W.: Function theory in the unit ball of ? n . Berlin, Heidelberg New York: Springer 1980 · Zbl 0495.32001
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