Fully Nonlinear Equations in Complex Geometry

复杂几何中的完全非线性方程

• 批准号: 1105786
• 负责人: Qun Li
• 金额: $ 10.57万
• 依托单位: Wright State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105786&HistoricalAwards=false
• 关键词: Fully; Nonlinear; Equations; Complex; Geometry

The research proposed aims to develop the interactions between fully nonlinear partial differential equations, complex geometry, and several complex variables. The PI plans to study some fundamental problems in three inter-related areas: Constant rank arguments in complex variables; totally real submanifolds of complex manifolds; and fully nonlinear equations in Hermitian manifolds. A very important theme in the area of geometric analysis is that understanding the solutions to some partial differential equations in terms of metrics, curvatures, or other geometric forms can be used to obtain more information about the geometry and topology of the manifolds. Under this consideration, in the study of complex geometry, the class of plurisubharmonic functions plays a crucial role. In her previous works, the PI has obtained, along with some important applications, a general constant rank result about the complex Hessian matrix of the solution to certain partial differential equations, which can be viewed as a refined statement of plurisubharmonicity. Along this direction, further study could be focused on discovering more geometric properties by the powerful tool of the constant rank argument. In the second part of the project, the PI is determined to discuss some nice geometric structures of real submanifolds in complex settings by studying the Dirichlet problem of a homogeneous complex Monge-Ampere equation whose solution characterizes a smooth totally real submanifold. In a recent joint work with B.Guan, the PI has established an optimal regularity result for such solutions. We would discover more interesting properties of the rich geometry that the totally real submanifolds provide us in this project. The last part of the proposed research is motivated from the first two and a natural continuation of a joint project with B. Guan about the complex Monge-Ampere equations in Hermitian manifolds. The PI with her collaborator would like to investigate some fully nonlinear equations under the general Hermitian setting, where the non-trivial torsion term gives us much trouble in the analysis. The proposed project links a wide range of active fields of mathematics, in particular, nonlinear partial differential equations, differential geometry, pluri-potential theory, and classical analysis, and furthermore in the broader fields of other subjects in sciences. Problems in this proposal arise naturally from our attempt to understand the behavior of solutions to nonlinear differential equations from geometry. The proposed research activity on geometry and regularity of nonlinear partial differential equations may bring in new and innovative progresses and lead to other significant geometric applications.

该研究旨在开发完全非线性偏微分方程、复杂几何和多个复杂变量之间的相互作用。 PI 计划研究三个相互关联领域的一些基本问题： 复杂变量中的常秩参数；复流形的全实子流形；以及厄米流形中的完全非线性方程。几何分析领域的一个非常重要的主题是，理解一些偏微分方程在度量、曲率或其他几何形式方面的解可以用来获得有关流形几何和拓扑的更多信息。在此考虑下，在复几何的研究中，多次调和函数类起着至关重要的作用。在她之前的工作中，PI 已经获得了关于某些偏微分方程解的复 Hessian 矩阵的一般常秩结果以及一些重要的应用，这可以被视为多次调和性的精化陈述。沿着这个方向，进一步的研究可以集中于通过常秩论证这个强大的工具发现更多的几何性质。在该项目的第二部分中，PI 决心通过研究齐次复 Monge-Ampere 方程的狄利克雷问题（其解表征了平滑的全实子流形）来讨论复杂环境中实子流形的一些良好几何结构。在最近与 B.Guan 的合作中，PI 为此类解决方案建立了最佳规律性结果。我们将在这个项目中发现完全真实的子流形为我们提供的丰富几何形状的更多有趣属性。拟议研究的最后一部分是由前两个部分激发的，也是与 B.guan 的一个关于厄米流形中复杂 Monge-Ampere 方程的联合项目的自然延续。 PI 和她的合作者希望研究一般 Hermitian 设置下的一些完全非线性方程，其中非平凡的扭转项给我们的分析带来了很多麻烦。拟议的项目连接了广泛的活跃数学领域，特别是非线性偏微分方程、微分几何、多势理论和经典分析，以及其他科学学科的更广泛领域。这个提议中的问题自然产生于我们试图理解几何中非线性微分方程解的行为。所提出的关于非线性偏微分方程的几何和正则性的研究活动可能会带来新的和创新性的进展，并导致其他重要的几何应用。