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

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).

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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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Correspondence to Nabil Ourimi.

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Supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

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Ourimi, N. Extension of CR maps between real-analytic hypersurfaces of different dimensions. Collect. Math. 66, 285–295 (2015). https://doi.org/10.1007/s13348-014-0115-x

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Keywords

  • Analytic sets
  • CR maps
  • Holomorphic correspondences
  • Segre varieties

Mathematics Subject Classification (2010)

  • 32H02
  • 32H40
  • 32H35
  • 53C15