Fridman, B. L. An approximate Riemann mapping theorem in \({\mathbb{C}}^ n\). (English) Zbl 0579.32041 Math. Ann. 275, 49-55 (1986). 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. Cited in 3 Documents 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 Keywords:unit ball; biholomorphic imbeddings; approximate Riemann mapping theorem in \({\mathbb{C}}^ n\) PDFBibTeX XMLCite \textit{B. L. Fridman}, Math. Ann. 275, 49--55 (1986; Zbl 0579.32041) Full Text: DOI EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.