Mathematical Sciences: Mapping Problems in Complex Analysis

数学科学：复分析中的映射问题

• 批准号: 8922810
• 负责人: Steven Bell
• 金额: $ 13.23万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1994-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8922810&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Mapping; Problems; Complex

Several important mathematical topics will be addressed in this project. Its primary focus is the theory of several complex variables with emphasis on (a) mappings between domains in spaces of the same or different dimensions, (b) extension to the boundary of such maps and (c) the resulting regularity one can expect. The basis for this work can be traced to the classical Riemann Mapping Theorem in one complex variable which states that any two finite and simply connected regions can be transformed into one another by univalent analytic mappings. The same question in several variables has two formulations. This is due to the observation that even elementary domains such as spheres and polydiscs cannot be transformed analytically into one another. The work considers those domains which can be analytically mapped onto one another in a one-one fashion and those which effectively can be mapped onto one another in a (more or less) finite-to-one-manner. The latter functions are called proper mappings. The research is concerned with questions of the following type. If a proper map transforms one domain into another, what conditions on the boundary or the map ensure that the map can be extended to the boundary? If an extension exists, what is its degree of smoothness? Considerable progress has been made on these questions when the two domains have curved and very regular boundaries. The object now is to reduce the assumptions on the boundaries and apply new results of complex analysis, geometry and partial differential equations to develop the theory to its fullest extent. The main problem to be addressed in the near term is considerable in itself: one must show that a continuous map between certain manifolds is actually infinitely differentiable. Results of this type are already known, but not enough to suggest a path to the best possible conclusions.

几个重要的数学主题将在 这个项目 它的主要焦点是几个理论 复变量，重点是（a）域之间的映射 在相同或不同维度的空间中，（B）扩展到 这类映射的边界和（c）由此产生的正则性 可以期待。 这项工作的基础可以追溯到古典 黎曼映射定理在一个复杂的变量，其中规定， 任意两个有限的单连通区域可以变换为 通过单叶解析映射相互转化。 相同的 多变量问题有两种提法。 这是由于 即使是像球体这样的基本域 多圆盘不能解析地转化为一个 另 这项工作考虑了那些可以 以一对一的方式分析映射到另一个上， 这些可以有效地映射到另一个（更多） 或更少）有限对一方式。 后面的函数称为 正确的映射 研究涉及以下问题 类型. 如果一个合适的映射将一个域转换为另一个域， 边界或地图上的条件确保地图可以 延伸到边界？ 如果存在一个扩展，它的 平滑度？ 在以下方面取得了相当大的进展： 当两个区域都是弯曲的并且非常规则时， 边界 现在的目标是减少对 边界并应用复杂分析、几何图形 和偏微分方程来发展这个理论， 最大程度。 近期要解决的主要问题 项本身是相当大的：必须表明，一个连续的 某些流形之间的映射实际上是无限的 可微的 这种类型的结果是已知的，但不是 足以让我们找到一条通往最佳结论的道路。