L2 Holomorphic Functions on Non-Compact Manifolds and Related Topics

Shubin提出研究带边界的非紧流形上的L^2全纯函数和L^2 Dolbeault上同调。他将使用冯·诺依曼代数技术进一步发展L^2奥卡-格劳特 具有伪凸或q-凸边界bM的复流形M的理论 设李群G在M上的全纯作用具有紧商M/G. 他计划研究M上的非平凡L^2全纯函数对于具体李群G的存在性， 在M上自由地具有紧商M/G。 这样一来， 可以描述群G的类，保证了非平凡L^2全纯函数的存在性。 如果没有集体行动， 也可以用来寻找保证非平凡L^2全纯函数存在的几何条件。 从多个复变量出发的技术类似于描述宇宙学模型（通过扭曲）和自由边界问题的线性和非线性偏微分方程中出现的技术， 赫姆霍兹和基尔霍夫。 对这些模型的研究可以导致 热核反应的重要发现，这是我们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.