ON AN n-MANIFOLD IN C NEAR AN ELLIPTIC COMPLEX TANGENT
ON AN n-MANIFOLD IN C NEAR AN ELLIPTIC COMPLEX TANGENT
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作者：
Xiaojun Huang
在本文中，我们将讨论实n维流形M在C中的局部双全纯性质。在一般的点上，这样的流形基本上具有标准R在C中的性质。然而，在复切线附近，考虑可能要复杂得多，流形可能获得非平凡的全纯局部壳和许多其他双全纯不变量。对这类问题的研究最早是在E.Bishop[BIS]的一篇著名论文中进行的，在该论文中，对于每个充分非退化的复正切，他附加了一个双全纯不变量λ，称为Bishop不变量。当复切线为椭圆时，即当0≤λ<；12时(更精确的定义见§2)，他证明了存在边界在M上的复解析圆盘族，这些圆盘族向下收缩到M中具有复切线的点的轨迹。特别地，利用众所周知的连续性原理，可以看到这类流形的像M̃包含在流形的全纯壳中。当时，他询问M̃是否准确地给出了M的局部全纯壳，以及所附圆盘的某些唯一性性质。他还提出了在这样复杂的切线附近确定M̃的精细结构的问题。后来，又出现了一系列关于M-̃C情形下M-⊂的光滑性的文章。这里我们要特别提到由Kenig-Webster在他们的深入工作[KW1]中证明的著名的定理：全纯M-̃在椭圆复切线附近的局部壳是一个光滑的⊂超曲面，其中M-Levi C是其光滑边界的一部分。在Moser-Webster[MW]的另一篇重要文章中，系统的范式理论被用来理解当M是实解析的情况下M的局部双全纯不变量。当毕晓普不变量λ6=0时，他们的方法适用于任何维度，甚至适用于某些双曲型复切线；但当λ=0时，它对复切线无效。其中，他们证明了M可以被双全纯映射到仿射空间
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