Fully Nonlinear Equations in Complex Geometry

这项研究旨在发展完全非线性偏微分方程组、复杂几何和多个复变量之间的相互作用。PI计划研究三个相互关联的领域中的一些基本问题：复变量中的常数阶变元；复流形的全实子流形；以及厄米特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，了解一些偏微分方程解的度量、曲率或其他几何形式，可以用来获得关于流形的几何和拓扑的更多信息。在这种情况下，在复杂几何的研究中，多次调和函数类起着至关重要的作用。在她以前的工作中，PI得到了关于某些偏微分方程解的复海森矩阵的一个一般常数秩结果，并得到了一些重要的应用，这可以看作是多重次谐和性的一个精化陈述。沿着这个方向，进一步的研究可以集中在发现更多的几何性质上，利用常数秩论这一强大的工具。在项目的第二部分，通过研究齐次复Monge-Ampere方程的Dirichlet问题，确定了PI来讨论复数环境下实子流形的一些良好的几何结构，其解的特征是光滑的全实子流形。在最近与B.Guan的合作中，PI建立了此类解的最优正则性结果。在这个项目中，我们将发现全实子流形为我们提供的丰富几何的更多有趣性质。这项研究的最后一部分是由前两部分和与B.Guan关于Hermitian流形上的复Monge-Ampere方程的联合项目的自然延续而来的。PI和她的合作者想要研究一般厄米特环境下的一些完全非线性方程，其中非平凡的挠率项给我们的分析带来了很大的麻烦。拟议的项目将广泛的数学活跃领域联系在一起，特别是非线性偏微分方程式、微分几何、多势理论和经典分析，以及更广泛的科学其他学科领域。由于我们试图从几何上理解非线性微分方程解的行为，所以这个建议中的问题自然会出现。所提出的关于非线性偏微分方程几何和正则性的研究活动可能会带来新的和创新的进展，并导致其他重要的几何应用。

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.