• Analytic sets and extension of holomorphic maps of positive codimension

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      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

       

      MARYAM AL-TOWAILB1 NABIL OURIMI1

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

       
  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.