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Analytic sets and extension of holomorphic maps of positive codimension

Article

Abstract

Let D, \(D'\) be arbitrary domains in \({\mathbb C}^n\) and \({\mathbb C}^N\) respectively, \(1<n\le N\), 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 \(\mathrm{cl}_{f}(M)\) does not intersect \(D'\). It is proved that if the cluster set \(\mathrm{cl}_{f}(p)\) 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 \(\partial D\).

Keywords

Analytic sets Holomorphic correspondences Segre varieties 

2000 Mathematics Subject Classification

32H40 32H35 32B15 

Notes

Acknowledgements

This work is supported by King Saud University, Deanship of Scientific Research, College of Science Research Center. The authors would like to thank Prof. R Shafikov for helpful discussions on the paper [21] and the referee for his/her valuable remarks.

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsKing Saud UniversityRiyadhSaudi Arabia

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