Mathematical Sciences: Several Complex Variables and PartialDifferential Equations

将对在这方面自然产生的几个复变量和某些偏微分方程式领域中出现的数学问题进行研究。总共有五个领域将得到治疗。第一个涉及类似于经典反射原理的反射原理。给出了一个定义在曲面一侧上的全纯映射，该映射定义在由多个复变量组成的空间中，并希望将该映射扩展到曲面上的一个完整邻域。最近在二维复数维上的结果完全解决了非Levi平坦超曲面的问题。这一条件在更高的维度上是不够的，将努力在表面上找到必要的附加条件，以确保反射原理。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曲面到另一个曲面的全纯映射联系在一起。要解决的两个主要问题涉及映射本质上有限的条件和确定条件，在这些条件下，一个给定的超曲面是否可以全纯映射到另一个给定的超曲面(以某种非奇异的方式)。还将研究定义在一般流形上的函数的全纯扩张，或者等价地：例如，当域是楔形且边界是该楔形的边缘时，能否识别沿区域边界扇区的全纯函数的限制？众所周知，在一定的限制条件下，伪凸光滑超曲面之间的CR-映射实际上是无穷可微的。在流形和映射的最弱可能条件下，工作仍在继续，以获得关于此类映射的光滑性信息。在过去一年中，在这方面取得了相当大的进展。这项研究的最后一个主题涉及到一个众所周知的结果的边界模拟，即在区域的内点处消失到无穷级的全纯函数在整个点的连通分支中消失。这不一定是在边界点上的情况，尽管在一大类函数的边界点上也有同样的现象。这将是该项目的一个目标，即确定这种“独特延续”的性质可以发生的确切条件。

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.