Mathematical Sciences: Analytic and Geometric Function Theory

该项目的工作将集中于下列方面出现的问题： 多复变量理论，多能理论， 多变量、特征类中的映射属性 并具有球面中紧致极小曲面的刚性。 研究溶液的稳定性， 复杂的Monge-Ampere算子的Dirichlet问题将是 贯彻 其中包括确定 复次调和函数，它构成了正确的域， 这个操作员。 稳定性问题的核心是 解决方案的边界数据和不均匀的一部分， 操作符. 解决方案收敛的程度， 相关元素的融合仍然是开放的。 关于映射函数的问题来自前面的 结果表明，足够高的无穷小边界 全纯映射的切触迫使映射是双全纯的。 工作将在自然扩展这项工作，以找到n- 多元点Pick-Nevanlinna不等式 线性映射多元刚性条件建立 一部分是身份。 极小曲面与常平均曲面的研究 曲率必然与对谐波的理解有关 地图 最近的工作表明，共形结构的集合 固定拓扑类型，可以实现为嵌入式 三球面中的极小曲面是紧的。 微创 浸没表面的数量是有限的。 工作将在 确定这是否可以简化为唯一性。 该研究对非线性偏微分方程具有应用价值 微分方程、微分几何和高等数学 维势理论

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.