Mathematical Sciences: Several Complex Variables and PartialDifferential Equations

工作将在几个复杂变量和某些偏微分方程领域中出现的数学问题上进行，这些问题在这种情况下自然出现。总共有五个领域将得到治疗。第一个是类似于经典原理的反射原理。给定一个定义在若干复变量空间内曲面一侧的全纯映射，并希望将该映射扩展到整个曲面的全邻域。最近在两个复杂维度上的结果完全解决了非列维平面超曲面的问题。这个条件在更高的维度上是不充分的，我们将努力在表面上找到必要的附加条件来保证反射原理。第二个研究方向是关于超曲面之间的全纯映射的性质，这些超曲面通过从一个到另一个的全纯映射相关联。要解决的两个主要问题涉及映射本质上是有限的条件和确定一个给定的超曲面是否可以全纯映射到另一个（以某种非奇异方式）的条件。我们还将研究定义在泛流形上的函数的全纯扩展，或者等价地：我们能否在一个区域边界的扇形上识别全纯函数的限制，例如，当这个区域是一个楔形，而边界是这个楔形的边缘？在一定的限制条件下，伪凸光滑超曲面之间的cr -映射实际上是无限可微的。在流形和映射的最弱条件下，正在继续努力获得这种映射的平滑信息。过去一年在这方面取得了相当大的进展。本研究的最后一个主题涉及一个众所周知的结果的边界模拟，即一个全纯函数在一个域的内点上消失到无限阶，在该点的整个连接分量上消失。在边界点上不一定是这种情况，尽管在一大类函数的边界点上也有同样的现象。这个项目的目标之一是确定这种“独特延续”属性发生的精确条件。

Work will be done on mathematical problems arising in the field of several complex variables and certain partial differential equations which occur naturally in this context. In all, five areas will be treated. The first concerns a reflection principle analogous to the classical one. One is given a holomorphic mapping defined on one side of a surface within the space of several complex variables and would like to extend the map across the surface to a full neighborhood. Recent results in two complex dimensions completely resolve the question for non- Levi-flat hypersurfaces. This condition is not sufficient in higher dimensions and efforts will be made to find the additional conditions necessary on the surface to ensure the reflection principle. A second line of investigation concerns properties of holomorphic mappings between hypersurfaces which are related by means of a holomorphic map from one into the other. Two main questions to be addressed concern conditions in which the map is essentially finite and the determination of conditions under which one can decide whether one given hypersurface can be holomorphically mapped into another (in some nonsingular fashion). Work will also be done on holomorphic extensions of functions defined on generic manifolds or, equivalently: can one identify restrictions of holomorphic functions along sector of the boundary of a domain, for example, when the domain is a wedge and the boundary is the edge of that wedge? It is known that under certain restrictive conditions CR-mappings between pseudoconvex smooth hypersurfaces which are diffeomorphisms are actually infinitely differentiable. Work is continuing in an effort to obtain smoothness information on such mappings under the weakest possible conditions, both on the manifolds and the mappings. Considerable progress has been made during the past year in this regard. The final theme of this research concerns a boundary analogue to the well-known result that a holomorphic function which vanishes to infinite order at an interior point of a domain vanishes throughout the connected component of the point. This need not be the case at a boundary point, although the same phenomenon obtains at boundary points for a large class of functions. It will be one objective of this project to determine precise conditions where this property of "unique continuation" can occur.