Mathematical Sciences: The Leviforms and CR Extension

这个项目将继续围绕多复变量和复几何理论来研究偏微分方程组和几何问题。我们将研究全纯函数对子流形的限制。这些函数总是满足某些方程，称为切向柯西黎曼方程。本课题的研究涉及一个逆问题：复欧氏空间子流形上的什么局部几何条件能保证每个满足切向柯西黎曼方程的函数都是全纯函数的局部限制？在研究这一基本问题时，将运用复杂几何和分析的方法。几何条件由由Levi形式度量的曲率条件和子流形上的扇形性质条件来描述。利用解析圆盘技术和FBI变换来构造给定CR函数的延拓。我们还将利用解析圆盘技术证明复欧氏空间中边界具有某种弱凸性的区域上的Hardy Littlewood极大函数型估计。这项研究的结果有望应用于偏微分方程组理论和微分几何。

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.