Mathematical Sciences: Analytic and Geometric Function Theory

这个项目的主要研究领域是研究多复变量中的位势理论，以及与多复变量空间中的流形相关的复微分几何。位势理论的工作涉及分析作为由复Monge-Ampere算子给出的偏微分方程解产生的多重亚调和函数。多次调和函数是位势理论的基本构件。它们是作为Monge-Ampere方程的解出现的。然而，关于Monge-Ampere算符的性质仍有许多悬而未决的问题，这些问题阻碍了多亚谐理论的发展。其中包括确定多次调和函数是否构成Monge-Ampere算子的自然域，以及当强迫项为Borel测度时，建立非齐次Monge-Ampere方程解的有界性。还将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。确定了某些在严格伪凸域上有限的重正化特征类。它们在证明Kahler-Einstein曲面的Chern形式是非负的(2，2)形式方面是有效的。由此引出了一个值得注意的事实，即这种陈形式的积分可以分解成只依赖于边界拓扑的一部分。目前正在进行确定所有重整化特征类的集合的工作，即有限的Kahler-Einstein度规的曲率张量的Chern-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.