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

• Autores: Nabil Ourimi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 2, 2015, págs. 285-295
• Idioma: inglés
• DOI: 10.1007/s13348-014-0115-x
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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).