zbMATH — the first resource for mathematics

Relative embeddings of discs into convex domains. (English) Zbl 0681.32019
Let \(\Omega\) be a domain in \({\mathbb{C}}^ n\). \(S\subset \Omega\) is called a discrete subset of \(\Omega\) if S has no limit points in \(\Omega\). The author proves a number of interesting theorems concerning the existence of holomorphically embedded discs containing such sets. Denote by \({\mathbb{D}}\) the open unit disc in the complex plane \({\mathbb{C}}.\)
Theorem 1. Let \(\Omega\) be a convex domain in \({\mathbb{C}}^ N\), \(N\geq 3\). Every discrete subset of \(\Omega\) is contained in a holomorphically embedded disc, i.e. in a submanifold of \(\Omega\) of the form F(\({\mathbb{D}})\), where F: \({\mathbb{D}}\to \Omega\) is a holomorphic embedding.
For \(N=2\) the result is weaker: in this case only the existence of a variety \(V=F({\mathbb{D}})\) is guaranteed, V containing the discrete subset and F: \({\mathbb{D}}\to \Omega\) being a proper holomorphic immersion.
Theorem 3. Let \(\Omega\) be a bounded strictly convex domain. Then the map F in the theorems above can be chosen to extend continuously to the closure \({\bar {\mathbb{D}}}.\)
By choosing a discrete subset of \(\Omega\) the closure of which contains the boundary \(\partial \Omega\) the following interesting corollary is obtained.
Corollary. Let \(\Omega \subset {\mathbb{C}}^ N\) be a bounded strictly convex domain. There is a continuous map F: \({\bar {\mathbb{D}}}\to {\bar \Omega}\), holomorphic in \({\mathbb{D}}\) and such that \(F(\partial {\mathbb{D}})=\partial \Omega\).
Reviewer: B.Jöricke

32H99 Holomorphic mappings and correspondences
32E35 Global boundary behavior of holomorphic functions of several complex variables
Full Text: DOI EuDML
[1] Acquistapace, E., Broglia, F., Tognoli, A.: A relative embedding theorem for Stein spaces. Ann. Sc. Norm. Super. Pisa, C1. Sci., IV Ser.2, 507-522 (1975) · Zbl 0313.32020
[2] Alexander, H.: Explicit embedding of the (punctured) disc intoC 2. Comment. Math. Helv.52, 539-544 (1977) · Zbl 0376.32011
[3] Bingener, J., Flenner, H.: Steinsche Räume zu vorgegebenen Singularitäten. Arch. Math.32, 34-37 (1979) · Zbl 0394.32003
[4] Bremermann, H.J.: Complex convexity. Trans. Am. Math. Soc.82, 17-51 (1956) · Zbl 0070.30402
[5] ?irka, E.M.: Complex analytic sets Nauka, Moscow (1985) (in russian)
[6] Forstneri?, F.: Polynomially convex hulls with piecewise smooth boundaries. Math. Ann.276, 97-104 (1986) · Zbl 0596.32024
[7] Globevnik, J.: Discs in the ball containing given discrete sets. Math. Ann.281, 87-96 (1988) · Zbl 0668.32026
[8] Globevnik, J.: Boundary interpolation and proper holomorphic maps from the disc to the ball. Math. Z.198, 143-150 (1988) · Zbl 0652.32015
[9] Globevnik, J., Stout, E.L.: The ends of varieties. Am. J. Math.108, 1355-1410 (1986) · Zbl 0678.32005
[10] Guillemin, V., Pollack, A.: Differential topology. Englewood Cliffs N.J.: Prentice-Hall, 1974 · Zbl 0361.57001
[11] Hoffman, K.: Banach spaces of analytic functions. Englewood Cliffs, N.J.: Prentice-Hall, 1962 · Zbl 0117.34001
[12] Hörmander, L.: An introduction to complex analysis in several variables. Amsterdam: North-Holland 1973 · Zbl 0271.32001
[13] Krantz, S.G.: Function theory of several complex variables. New York: Wiley-Interscience, 1982 · Zbl 0471.32008
[14] Narasimhan, R.: Imbedding of holomorphically complete complex spaces. Am. J. Math.82, 917-934 (1960) · Zbl 0104.05402
[15] Pommerenke, C.: Univalent functions. (Math. Lehrb., Bd. 25). Göttingen: Vanderhoeck and Rupprecht 1975 · Zbl 0298.30014
[16] Rudin, W.: Real and complex analysis. New York: McGraw-Hill, 1966 · Zbl 0142.01701
[17] Stout, E.L.: The bounded extension problem. The case of discs in polydiscs. J. Anal. Math.28, 239-254 (1975) · Zbl 0317.32015
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.