Mathematical Sciences: Several Complex Variables and PartialDifferential Equations

工作将做的数学问题所产生的 多复变域和某些偏微分域 在这种情况下自然发生的微分方程。 在 共有五个地区会接受治疗。 第一个问题涉及到 原理类似于经典原理。 一个人被给予一个 定义在曲面一侧的全纯映射， 空间的几个复杂的变量，并希望扩大 映射到整个区域。 最近的结果 两个复杂的维度完全解决了非 Levi平坦超曲面但这个条件并不足以 更高的维度和努力将被用来寻找额外的 表面上的必要条件，以确保反射 原则 第二条调查路线涉及 相关超曲面之间的全纯映射 从一个到另一个的全纯映射。 两个主要 要解决的问题涉及地图被 本质有限和条件的确定 它可以决定一个给定的超曲面是否可以 全纯映射到另一个（在一些非奇异 时尚）。 工作也将完成全纯扩展 函数定义在通用流形上，或者等价地：可以一个 全纯函数沿着扇区限制识别 域的边界，例如，当域为楔形时 边界就是这个楔形的边缘 众所周知，在某些限制条件下， 伪凸光滑超曲面之间的CR-映射 实际上，超同态是无穷可微的。 工作是 继续努力获得关于这种平滑度的信息， 映射在最弱的可能条件下，无论是在 歧管和 映射。 2000年期间取得了相当大的进展， 去年在这方面。 本研究的最后一个主题涉及一个边界 类似于著名的结果，全纯函数 在区域的内点消失到无穷阶 在该点的整个连通分量中消失。 这 边界点的情况不一定如此，尽管相同 现象获得在边界点为一大类 功能协调发展的 本项目的目标之一是确定 精确的条件下，这个属性的“唯一连续” 可能发生。

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.