## 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

### MSC:

 32H99 Holomorphic mappings and correspondences 32E35 Global boundary behavior of holomorphic functions of several complex variables
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### References:

 [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
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