Fully Nonlinear Equations in Complex Geometry

该研究旨在发展完全非线性偏微分方程、复杂几何和多个复变量之间的相互作用。PI计划在三个相互关联的领域研究一些基本问题：复变量中的常秩参数;复流形的全真实的子流形;以及Hermitian流形中的全非线性方程。在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的度量，曲率或其他几何形式的解决方案可以用来获得更多关于流形的几何和拓扑结构的信息。在这种情况下，复几何的研究中，多重次调和函数类起着至关重要的作用。在她以前的工作中，PI已经获得了，沿着一些重要的应用，一个一般的常秩结果的复杂的海森矩阵的解决方案，某些偏微分方程，这可以被视为一个精炼的陈述pluissubharmonicity。沿着这一方向，进一步的研究可以集中在利用常秩论证这一强有力的工具来发现更多的几何性质上。在项目的第二部分中，PI被确定为通过研究齐次复Monge-Ampere方程的Dirichlet问题来讨论复背景下真实的子流形的一些好的几何结构，该方程的解表征光滑的全真实的子流形。在最近与B.Guan的联合工作中，PI已经为这样的解决方案建立了最佳正则性结果。在这个项目中，我们将发现全真实的子流形提供给我们的丰富几何的更有趣的性质。建议研究的最后一部分是从前两个和一个自然的延续与B的联合项目的动机。关晓彤关于厄米流形中的复杂Monge-Ampere方程。PI和她的合作者想研究一些完全非线性方程下的一般厄米特设置，其中的非平凡扭转项给我们带来了很多麻烦的分析。该项目将广泛的数学活跃领域联系起来，特别是非线性偏微分方程，微分几何，多能理论和经典分析，以及更广泛的其他科学学科领域。这个建议中的问题自然出现在我们试图从几何角度理解非线性微分方程解的行为时。拟议的非线性偏微分方程几何和正则性研究活动可能会带来新的创新进展，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.