ON AN n-MANIFOLD IN C NEAR AN ELLIPTIC COMPLEX TANGENT

In this paper, we will be concerned with the local biholomorphic properties of a real n-manifold M in C. At a generic point, such a manifold basically has the nature of the standard R in C. Near a complex tangent, however, the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants. The study of such a problem was first carried out in a celebrated paper of E. Bishop [BIS] where, for each sufficiently non-degenerate complex tangent, he attached a biholomorphic invariant λ, called the Bishop invariant. When the complex tangent is elliptic, i.e., when 0 ≤ λ < 12 (for a more precise definition, see §2), he showed the existence of families of complex analytic disks with boundary on M that shrink down to the locus of points in M with complex tangents. In particular, using the well-known continuity principle, one sees that the image M̃ of such families is contained in the holomorphic hull of the manifold. At the time, he asked whether M̃ gives precisely the local holomorphic hull of M , as well as certain uniqueness properties of the attached disks. He also proposed the problem of determining the fine structure of M̃ near such complex tangents. Later, there appeared a sequence of papers concerning the smooth character of M̃ in case M ⊂ C. Here we would like to mention, in particular, the famous theorem proved by Kenig-Webster in their deep work [KW1] which states that the local hull of holomorphy M̃ near an elliptic complex tangent is a smooth Levi flat hypersurface with M ⊂ C as part of its smooth boundary. In another important paper of Moser-Webster [MW], a systematic normal form theory was employed for the understanding of the local biholomorphic invariants of M in case M is real analytic. When the Bishop invariant λ 6= 0, their method works in any dimension and even for some hyperbolic complex tangents; but it breaks down for complex tangents with λ = 0. Among other things, they showed that M can be biholomorphically mapped into the affine space