L2 Holomorphic Functions on Non-Compact Manifolds and Related Topics

非紧流形上的 L2 全纯函数及相关主题

• 批准号: 9706038
• 负责人: Mikhail Shubin
• 金额: --
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706038&HistoricalAwards=false
• 关键词: L2; Holomorphic; Functions; Non; Compact

Shubin proposes to study L^2 holomorphic functions and L^2 Dolbeault cohomology on non-compact manifolds with boundary. He will use Von Neumann algebra techniques to further develop the L^2 Oka-Grauert theory on complex manifolds M with pseudoconvex or q-convex boundary bM admitting a holomorphic action of a Lie group G on M with a compact quotient M/G. He plans to investigate existence of non-trivial L^2 holomorphic functions on M for concrete Lie groups G which act freely on M with compact quotient M/G. In this way it might be possible to describe the class of groups G for which the existence of non-trivial L^2 holomorphic functions is guaranteed. The situation when there is no group action will be also considered with the purpose of finding geometric conditions which guarantee existence of non-trivial L^2 holomorphic functions. Techniques from several complex variables are similar to those appearing in linear and non-linear partial differential equations that describe cosmological models (through twistors) and free boundary problems, providing models for plasma in tokamaks by methods of Helmholtz and Kirchhoff. Investigation of these models can lead to important discoveries in thermonuclear reactions which are our main hope for the future source of cheap energy.

Shubin提出了研究L^2的全态函数，并在具有边界的非紧凑型歧管上研究L^2 Dolbeault的同谋。他将使用von Neumann代数技术来进一步开发与Pseudoconvex或Q-Convex边界BM的复杂歧管上的L^2 OKA-GRAUERT理论，该理论BM承认Lie Glop g在M上具有紧凑的商M/G的M lie Glove g。 他计划调查M lie lie组的非平凡l^2全态函数G组G组G，该谎言g群是在M/g上自由作用于M的M。 通过这种方式，可以保证可以保证存在非平凡的L^2骨术功能的G组类别的类别。 还将考虑没有群体行动的情况，目的是找到几何条件，以保证存在非平凡的l^2全态函数。 来自几个复杂变量的技术类似于在线性和非线性部分微分方程中出现的技术，这些方程描述了宇宙学模型（通过扭曲器）和自由边界问题，通过Helmholtz和Kirchhoff的方法为Tokamaks中的血浆提供了模型。 对这些模型的研究可能会导致热核反应的重要发现，这是我们对未来廉价能源的来源的主要希望。