MARYAM AL-TOWAILB
Articles written in Proceedings – Mathematical Sciences
Volume 128 Issue 3 June 2018 Article ID 0033 Research Article
Analytic sets and extension of holomorphic maps of positive codimension
MARYAM AL-TOWAILB NABIL OURIMI
Let $D$, $D'$ be arbitrary domains in $\mathbb{C}^{n}$ and $\mathbb{C}^{N}$ respectively, 1 < $n \leq N$, both possibly unbounded and $M \subseteq \partial D$, $M' \subseteq \partial D'$ be open pieces of the boundaries. Suppose that $\partial D$ is smooth real-analytic and minimal in an open neighborhood of $\bar{M}$ and $\partial D'$ is smooth real-algebraic and minimal in an open neighborhood of $\bar{M}'$. Let $f : D \rightarrow D'$ be a holomorphic mapping such that the cluster set $\rm{cl}_{f}(M)$ does not intersect $D'$. It is proved that if the cluster set $\rm{cl}_{f}(p)$ of some point $p \in M$ contains some point $q \in M'$ and the graph of f extends as an analytic set to a neighborhood of $(p, q) \in \mathbb{C}^{n} \times \mathbb{C}^{N}$ , then $f$ extends as a holomorphic map to a dense subset of some neighborhood of $p$. If in addition, $M = \partial D$, $M' = \partial D'$ and $M'$ is compact, then $f$ extends holomorphically across an open dense subset of $\partial D$.
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