L2 Holomorphic Functions on Non-Compact Manifolds and Related Topics

舒斌提出研究带边界非紧流形上的L^2全纯函数和L^2 Dolbeault上同调。他将利用Von Neumann代数技巧进一步发展关于具有伪凸或q凸边界BM的复流形上的L^2 Oka-Grauert理论，该理论允许具有紧商M/G的李群G在M上的全纯作用。他计划研究具有紧商M/G的具体李群G在M上的非平凡L^2全纯函数的存在性。这样，就可能描述保证其非平凡全纯函数存在的群G。也将考虑不存在群作用的情形，目的是找到保证存在非平凡的L^2全纯函数的几何条件。来自几个复变量的技术类似于出现在描述宇宙学模型(通过扭曲)和自由边界问题的线性和非线性偏微分方程组中的技术，通过Helmholtz和Kirchhoff的方法提供托卡马克中的等离子体模型。对这些模型的研究可以导致在热核反应方面的重要发现，这是我们对未来廉价能源的主要希望。

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.