Mathematical Sciences: Analytic Discs in Several Complex Variables

数学科学：多个复变量的解析盘

• 批准号: 9204384
• 负责人: Alexander Tumanov
• 金额: $ 4.08万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1994-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204384&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analytic; Discs; Several

This award supports mathematical research on problems concerned primarily with the extension of functions and mappings defined on boundaries of domains in several complex variables. The boundaries are generalizations of surfaces, known as (real) manifolds. The idea of extending a function from the manifold to a holomorphic function on the domain is actually a boundary value problem in partial differential equations. Functions which can be extended must satisfy certain (Cauchy-Riemann) conditions on the manifold. It has recently been shown that all CR functions on a generic manifold extend over a wedge in a neighborhood of some point of the manifold. Work will now be done adapting newly developed techniques to establish the location of the wedge. Another research direction will be that of determining geometric conditions for extending CR functions over a neighborhood of the manifold. Analyticity or wedge-extendibility of a CR function on a CR manifold propagates along complex curves in the manifold. A natural consideration is also the propagation of CR extendibility. Work will be done in showing that CR extendibility is transported in parallel with respect to some connection on M. A basic tool in all the work is that of using analytic discs to localize the analysis. Work on mappings between CR manifolds will focus on continuous maps between analytic manifolds to determine the extent to which continuity actually implies that the mappings are holomorphic. Some initial smoothness assumptions will be made coupled with the use of the reflection principle on attached analytic discs.

这个奖项支持数学研究的问题 主要涉及函数和映射的扩展 定义在多个复变量的域的边界上。 边界是曲面的推广，称为（真实的） 流形将函数从流形扩展到 定义域上的全纯函数实际上是一个边值 偏微分方程中的一个问题 函数可以 必须满足某些（Cauchy-Riemann）条件， 歧管。 最近的研究表明，所有的CR功能都是在一个 一般流形在某个邻域中的楔上延伸 歧管的点。 现在的工作将是适应新的 开发了确定楔形物位置的技术。 另一个研究方向将是确定几何 条件扩展CR函数的邻域 歧管 CR上CR函数的解析性或楔可扩性 流形沿着流形中的复杂曲线传播。 一 自然的考虑也是CR的传播 可扩展性 工作将在显示CR 可扩展性是相对于一些 连接M。 所有工作中的一个基本工具是使用 分析盘以本地化分析。 CR流形之间的映射工作将集中在 解析流形之间的连续映射，以确定 连续性实际上意味着映射是 全纯的 将进行一些初始平滑假设 再加上利用反射原理， 分析盘