Mathematical Sciences: Analytic and Geometric Function Theory

这个项目的工作将集中在多复变量理论、多势理论、多变量映射性质、特征类以及球面上紧致极小曲面的刚性等问题上。研究复Monge-Ampere算子解的稳定性和Dirichlet问题。这包括确定构成该算子的正确区域的多重子调和函数的类别。稳定性问题的中心是解对边界数据和算子的非齐次部分的依赖性。当这些相关元素汇聚时，解决方案的汇聚程度仍然是开放的。关于映射函数的问题源于以前的结果，这些结果表明全纯映射的足够高的无穷小边界接触迫使该映射是双全纯的。这项工作的自然推广将被用来找出多个变量的n点Pick-Nevanlinna不等式，并在其线性部分为恒等式的映射上建立多元刚性条件。极小曲面和常平均曲率曲面的研究必然与对调和映射的理解相联系。最近的工作表明，可以实现为三球面中嵌入极小曲面的固定拓扑型共形结构集是紧的。最小浸没曲面的数量是有限的。将进行工作，以确定这是否可以降低到独特性。这项研究在非线性偏微分方程组、微分几何和高维势理论中都有应用。

Work on this project will concentrate on problems arising in the theory of several complex variables, pluripotential theory, mapping properties in several variables, characteristic classes and with the rigidity of compact minimal surfaces in spheres. Investigations into the stability of solutions and the Dirichlet problem for the complex Monge-Ampere operator will be carried out. These include determining the class of plurisubharmonic functions which make up the correct domain for this operator. Stability questions center on the dependence of solutions on both boundary data and the inhomogenous part of the operator. The extent to which solutions converge when these related elements converge is still open. Questions regarding mapping functions derive from earlier results which show that high enough infinitesimal boundary contact of holomorphic maps forces the map to be biholomorphic. Work will be done on natural extensions of this work to find n- point Pick-Nevanlinna inequalities in several variables and to establish multivariate rigidity conditions on maps whose linear part is the identity. Studies of minimal surfaces and surfaces of constant mean curvature are necessarily linked to the understanding of harmonic maps. Recent work has shown that the set of conformal structures of fixed topological type that can be realized as embedded minimal surfaces in the three-sphere is compact. Minimally immersed surfaces are finite in number. Work will be done in establishing whether or not this can be reduced to uniqueness. This research has application to nonlinear partial differential equations, differential geometry and higher dimensional potential theory.