Abstract
We consider a CR mapping
f
:
M
→
M
′
between real-analytic hypersurfaces of finite D’Angelo type in complex spaces
C
n
+
1
and
C
N
+
1
, respectively, that extends as a holomorphic correspondence to a neighborhood of some point
z
0
∈
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
).