## 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:

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