Nonvanishing of the Differential of Holomorphic Mappings at Boundary Points

Let M and M ′ be two smooth hypersurfaces in C. A smooth mapping h : M → M ′ is a CR mapping if its components are annihilated by the induced CauchyRiemann operator on M . Let p0 ∈ M and suppose that near p0, h is the restriction of a holomorphic mapping H defined on one side of M near p0 and smooth up to M . We shall say that h satisfies the Hopf lemma property at p0 if the component of H normal to M2 has a nonzero derviative at p0 in the normal direction to M1. The hypersurface M is said to be minimal at p0 ∈ M if there is no germ of a complex hypersurface contained in M through p0. Recall the theorem of Trépreau [T] that ifM is minimal at p0, then every CR function defined onM near p0 extends holomorphically to at least one side of M in C near p0. A stronger condition on a hypersurface M at a point p0 is that of essential finiteness (as defined in [BJT], [BR3], [DA2]). We will recall this definition in §1. We note here that if M is of D’Angelo finite type at p0 [DA1], then M is essentially finite at p0 (and hence minimal at p0). In this paper we prove a general result of the “Hopf lemma” type for CR mappings, with nonidentically vanishing Jacobians, between real hypersurfaces in C. Applications of this result to finiteness and holomorphic extendibility of such mappings are also given. The novelty here is that we make no assumption on the nonflatness of the mapping or its Jacobian, nor do we assume that the hypersurfaces are pseudoconvex or minimally convex.