Mathematical Sciences: Several Complex Variables and PartialDifferential Equations

数学科学：多个复变量和偏微分方程

• 批准号: 8901268
• 负责人: Linda Rothschild
• 金额: $ 21.5万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1994-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901268&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Several; Complex; Variables

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.

将研究在几个复数变量和在这种情况下自然出现的某些偏微分方程领域中出现的数学问题。 总共将处理五个区域。 第一个涉及类似于经典原理的反射原理。 给定一个在多个复杂变量的空间内的表面一侧定义的全纯映射，并且希望将该映射扩展到整个表面到完整的邻域。 最近在两个复杂维度上的结果完全解决了非李维平坦超曲面的问题。这个条件在更高维度上是不够的，我们将努力寻找表面上必要的附加条件以确保反射原理。 研究的第二条线涉及超曲面之间的全纯映射的属性，这些超曲面通过从一个到另一个的全纯映射进行关联。 要解决的两个主要问题涉及映射本质上有限的条件以及确定一个给定超曲面是否可以全纯映射到另一个超曲面（以某种非奇异方式）的条件。 还将研究泛型流形上定义的函数的全纯扩展，或者等效地：可以识别沿域边界扇区的全纯函数的限制，例如，当域是楔形且边界是那个楔子？ 众所周知，在某些限制条件下，伪凸光滑超曲面（微分同胚）之间的 CR 映射实际上是无限可微的。 正在继续努力在最弱的可能条件下获得此类映射的平滑信息，无论是流形还是映射。 去年，这方面取得了长足进展。 这项研究的最后一个主题涉及与众所周知的结果的边界模拟，即在域的内点处消失到无限阶的全纯函数在该点的整个连通分量中消失。 尽管在一大类函数的边界点处会出现相同的现象，但边界点处的情况不一定如此。 该项目的一个目标是确定这种“唯一延续”属性可以发生的精确条件。