Mathematical Sciences: Analytic Discs in Several Complex Variables

该奖项支持对主要与定义在几个复变量的区域边界上的函数和映射的扩展有关的问题的数学研究。边界是曲面的推广，称为(实)流形。将函数从流形延拓到区域上的全纯函数实际上是偏微分方程组中的边值问题。可以扩张的函数必须满足流形上的某些(柯西-黎曼)条件。最近证明了一般流形上的所有CR函数都延伸到流形上某一点邻域内的楔形上。现在将进行工作，采用新开发的技术来确定楔子的位置。另一个研究方向是确定在流形邻域上扩张CR函数的几何条件。CR流形上CR函数的解析性或楔形可扩性沿流形中的复杂曲线传播。一个自然的考虑因素也是CR可扩展性的传播。将做的工作是证明CR的可扩展性相对于M上的某些连接是并行传输的。所有工作中的一个基本工具是使用分析盘来定位分析。关于CR流形之间映射的工作将集中于解析流形之间的连续映射，以确定连续在多大程度上实际上意味着映射是全纯的。在附加的分析圆盘上，结合反射原理的使用，将作出一些初始的光滑性假设。

This award supports mathematical research on problems concerned primarily with the extension of functions and mappings defined on boundaries of domains in several complex variables. The boundaries are generalizations of surfaces, known as (real) manifolds. The idea of extending a function from the manifold to a holomorphic function on the domain is actually a boundary value problem in partial differential equations. Functions which can be extended must satisfy certain (Cauchy-Riemann) conditions on the manifold. It has recently been shown that all CR functions on a generic manifold extend over a wedge in a neighborhood of some point of the manifold. Work will now be done adapting newly developed techniques to establish the location of the wedge. Another research direction will be that of determining geometric conditions for extending CR functions over a neighborhood of the manifold. Analyticity or wedge-extendibility of a CR function on a CR manifold propagates along complex curves in the manifold. A natural consideration is also the propagation of CR extendibility. Work will be done in showing that CR extendibility is transported in parallel with respect to some connection on M. A basic tool in all the work is that of using analytic discs to localize the analysis. Work on mappings between CR manifolds will focus on continuous maps between analytic manifolds to determine the extent to which continuity actually implies that the mappings are holomorphic. Some initial smoothness assumptions will be made coupled with the use of the reflection principle on attached analytic discs.