×

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
PDF BibTeX XML Cite
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.