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

Nabil Ourimi

• We consider a CR mapping ?:?→?′ between real-analytic hypersurfaces of finite D’Angelo type in complex spaces ℂ?+1 and ℂ?+1 , respectively, that extends as a holomorphic correspondence to a neighborhood of some point ?0∈? and that ?′ is Levi-nondegenerate at ?′0=?(?0) . In this paper, we give sufficient conditions to extend ? as a holomorphic mapping across ?0 . In contrast with the equidimensional case, our result fails in general, when ?′ is Levi-degenerate at ?′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 ? where the rank of the mapping is maximal; equal to ?+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).