Mathematical Sciences: Analytic and Geometric Function Theory

这个项目的主要研究领域是研究几个复杂变量的势理论和与几个复杂变量空间中的流形相关的复杂微分几何。势理论的工作是分析由复蒙日-安培算符给出的偏微分方程的解所产生的多次谐波函数。多次谐波函数是势理论的基本组成部分。它们是蒙日-安培方程的解。然而，关于蒙日-安培算符的性质有许多悬而未决的问题，这些问题阻碍了多次谐波理论的发展。它们包括确定多次谐波函数是否构成蒙日-安培算子的自然域，以及当强迫项为Borel测度时建立非齐次蒙日-安培方程解的有界性。工作也将完成研究复杂流形及其边界的解析和几何性质，特别是柯西-黎曼流形。在严格伪凸域上，确定了一些有限的重正则化特征类。他们有效地证明了Kahler-Einstein曲面的Chern形式是非负的（2,2）形式。由此可以得出一个值得注意的事实，即这种陈氏形式的积分可以分解成只依赖于边界拓扑的部分。现在正在进行的工作是确定所有重归一化特征类的集合，即有限的Kahler-Einstein度量曲率张量的chen - weyl多项式的整体积分。

The primary areas of investigation in this project are studies of potential theory in several complex variables and complex differential geometry related to manifolds in the space of several complex variables. Work in potential theory concerns the analysis of plurisubharmonic functions arising as solutions of a partial differential equation given by the complex Monge-Ampere operator. The plurisubharmonic functions are the basic building blocks for potential theory. They arise as solutions of Monge-Ampere equations. However, there are many unanswered questions regarding the nature of the Monge-Ampere operator which impede development of a plurisubharmonic theory. They include determining whether or not the plurisubharmonic functions form the natural domains of Monge-Ampere operator and establishing the boundedness of solutions of the inhomogeneous Monge-Ampere equation when the forcing term is a Borel measure. Work will also be done studying the analytic and geometric properties of complex manifolds and their boundaries, specifically Cauchy-Riemann manifolds. Certain renormalized characteristic classes, finite on strictly pseudoconvex domains, have been identified. They have proved effective in showing that the Chern form for Kahler-Einstein surfaces is a nonnegative (2,2) form. From this follows the remarkable fact that the integral of this Chern form can be broken into a part only dependent on the boundary topology. Work is now proceeding to determine the set of all renormalized characteristic classes, that is, global integrals of Chern-Weyl polynomials of curvature tensors of Kahler-Einstein metrics which are finite.