Mathematical Sciences: CR Manifolds and Holomorphic Embeddings

首席研究员将分析的问题， 一类CR的全纯嵌入的存在性和正则性 真实的超曲面型流形。 这些流形 与在边界上诱导的相同的内部结构 复欧氏空间中光滑强伪凸域。 所应用方法的关键要素是Henkin解 切向柯西-黎曼方程的算子。 为 他将改善先前的估计并减少， 尽可能多地，他的大量衍生品损失 现有的论点。 全局嵌入的问题将是 进一步分析。 这项研究将使我们更深入地了解 复流形的边界概念 某些超定线性系统的性质，首先 阶偏微分方程。 平面上的一条无限的曲线， 浸入不是真实的线的示例 嵌入 嵌入总是设法避开自己。 全纯嵌入要求函数，它注入 将曲线或曲面导入目标流形，在 复杂的分析意义 研究CR或复形的定理 从标准上看，解析流形总是不可思议的 传统的真实的分析。 结果完成人 这个项目也不会有什么不同。

The principal investigator will analyze the problem of existence and regularity of holomorphic embeddings of certain CR manifolds of real hypersurface type. These are manifolds having the same internal structure as is induced on the boundary of a smooth, strongly pseudoconvex domain in complex Euclidean space. Key elements of the method to be applied are Henkin's solution operators for the tangential Cauchy-Riemann equations. For the local problem he will improve previous estimates and reduce, as much as possible, the considerable derivative loss in his existing arguments. The problem of global embeddings will be further analyzed. The study will lead to a deeper understanding of the concept of boundary of a complex manifold and of properties of certain over-determined systems of linear, first order, partial differential equations. An endless curve in the plane which wraps up on itself is an example of an immersion of the real line which is not an embedding. An embedding always manages to steer clear of itself. Holomorphic embeddings require that the function, which injects the curve or surface into the target manifold, be smooth in the complex analytic sense. Theorems from the study of CR or complex analytic manifolds have always seemed miraculous by the standards of traditional real analysis. The results to be completed by this project will be no different.