• Analytic sets and extension of holomorphic maps of positive codimension

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/128/03/0033

• # Keywords

Analytic sets; Holomorphic correspondences; Segre varieties

• # Abstract

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 $\partial D$.

• # Author Affiliations

1. Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019