Mathematical Sciences: The Leviforms and CR Extension

该项目将继续研究局部 微分方程与几何问题 多复变理论和复几何。 工作将完成的限制全纯函数 到子流形 这些函数总是满足 切向柯西黎曼方程 方程 对本项目进行的调查 涉及匡威的问题：什么局部几何条件 在复欧氏空间的子流形上，保证每个 满足切向柯西黎曼方程的函数是 全纯函数的局部约束 方法 复杂的几何和分析将应用于研究这一点， 根本问题。 几何条件描述为： 用Levi形式和扇形测量的曲率条件 子流形上的性质条件。 分析盘 技术和FBI变换将被用来构建 一个给定的CR函数的扩展。 还将在应用分析技术方面开展工作 证明哈代Littlewood极大函数型估计的圆片 关于复欧氏空间中边界具有 一些弱形式的凸性。 这项研究的结果是 预期将应用于部分理论 微分方程和微分几何。

This project will continue work on problems of partial differential equations and geometric questions centering on the theory of several complex variables and complex geometry. Work will be done on restrictions of holomorphic functions to submanifolds. These are functions which always satisfy certain equations known as the tangential Cauchy Riemann equations. The investigations to be carried out on this project involve the converse question: What local geometric conditions on a submanifold of complex Euclidean space guarantee that each function satisfying the tangential Cauchy Riemann equations is locally the restriction of holomorphic function? Methods of complex geometry and analysis will be applied in studying this fundamental question. The geometric conditions are described by curvature conditions as measured by the Levi forms and sector property conditions on the submanifold. The analytic disc technique and the FBI transform will be used to construct the extension of a given CR function. Work will also be done in applying the technique of analytic discs to prove Hardy Littlewood maximal function type estimates on domains in complex Euclidean space whose boundaries possess some weak form of convexity. Results of this research are expected to have application to the theory of partial differential equations and to differential geometry.