Collectanea Mathematica

, Volume 66, Issue 2, pp 285–295 | Cite as

Extension of CR maps between real-analytic hypersurfaces of different dimensions

Article
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Abstract

We consider a CR mapping \(f: M\rightarrow M'\) between real-analytic hypersurfaces of finite D’Angelo type in complex spaces \({\mathbb C}^{n+1}\) and \({\mathbb C}^{N+1}\), respectively, that extends as a holomorphic correspondence to a neighborhood of some point \(z_0\in M\) and that \(M'\) is Levi-nondegenerate at \(z_0'=f(z_0)\). In this paper, we give sufficient conditions to extend \(f\) as a holomorphic mapping across \(z_0\). In contrast with the equidimensional case, our result fails in general, when \(M'\) is Levi-degenerate at \(z_0'\). The proof uses the transversality of the mapping, which can be regarded as a type of Hopf’s lemma, the existence of points in \(M\) where the rank of the mapping is maximal; equal to \(n+1\) and the reflection principle in several variables. Related results were proved by Huang (Comm Partial Differ Equ 25:299–317, 2000); Pinchuk and Verma (Proc Am Math Soc 129(9):2623–2632, 2001); Diederich and Pinchuk (Doc Math 2:703–712, 1998); Diederich and Pinchuk (J Geom Anal 14(2):231–239, 2004) and Meylan et al. (Asian J Math 7(4):493–509, 2003).

Keywords

Analytic sets CR maps Holomorphic correspondences Segre varieties 

Mathematics Subject Classification (2010)

32H02 32H40 32H35 53C15 

Notes

Acknowledgments

The author would like to thank professor R. Shafikov for pointing out the Example 1 and the referee for his comments and suggestions which improved the paper greatly.

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Copyright information

© Universitat de Barcelona 2014

Authors and Affiliations

  1. 1.Department of mathematicKing Saud UniversityRiyadhKing Saudi Arabia

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