Fully Nonlinear Geometric Partial Differential Equations

这个项目涉及在微分几何，偏微分方程和理论物理的接口的基本问题的调查。自然界的基本定律是用数学语言来描述的，使用的是微分几何和偏微分方程的思想。理解微分方程解的行为是理解宇宙结构的重要组成部分。这个提议涉及一个有着几百年历史的数学对象，称为蒙日-安培方程。这些方程在几何学的研究中自然产生，与广义相对论中的爱因斯坦方程和弦理论中的赫尔-斯特罗明格系统密切相关。该项目的一个主要目标是开发分析技术，以研究这类方程的解的重要性质，这将导致对基本几何结构的深入理解。这项研究需要广泛的工具，从真实的和复杂的分析，以及代数和微分几何。这些问题的进展不仅对数学中的一些基本问题有所启发，而且在物理学和其他科学中也有应用。本项目将研究完全非线性偏微分方程与复几何之间的相互作用。特别是，研究者将研究复杂的Monge-Ampere型方程的可解性，这些方程是从弦理论中的广义Hull-Strominger系统的研究中推导出来的，它可以被视为非Kahler复流形上的Ricci平坦度量的推广。在他与Phong和Picard的联合工作的基础上，研究人员将开发新的分析技术，以进一步理解这种类型的方程，并为Fu和Yau提出的问题提供完整的答案。该项目的另一个主要目标是研究异常流，这是由研究者和他的合著者介绍的，旨在发展求解一般三维非Kahler Calabi-Yau流形上的Hull-Strominger系统的分析方法。在与Phong和Picard的联合工作中获得了流动解的短时存在性。研究者将研究这个流的长时间行为和收敛性，并将利用这个流研究Kahler流形上平衡锥和Kahler锥之间的关系。为了实现这些目标，研究人员将开发新的工具，用于研究非线性椭圆和抛物型偏微分方程，而不具有非线性性质。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

This project concerns investigations of fundamental problems at the interface of differential geometry, partial differential equations and theoretical physics. The fundamental laws of nature are described in the language of mathematics using ideas from differential geometry and partial differential equations. Understanding the behavior of solutions to differential equations is an important part of understanding the structure of the universe. This proposal concerns a centuries-old class of mathematical objects called Monge-Ampere equations. These equations arise naturally in the study of geometry and are closely related to Einstein's equation in general relativity and the Hull-Strominger system from string theory. A main goal of this project is to develop analytical techniques to investigate the important properties of the solutions to this type of equations, which will lead to deep understanding of the fundamental geometric structures. The study requires a broad range of tools from real and complex analysis, as well as algebraic and differential geometry. Progress on these questions will not only shed some light on some basic problems in mathematics, but will also have applications in physics and other sciences.This project will investigate the interaction between fully nonlinear partial differential equations and complex geometry. In particular, the investigator will study the solvability of complex Monge-Ampere type equations deduced from the study of a generalized Hull-Strominger system in string theory, which can be viewed as a generalization of Ricci-flat metrics on non-Kahler complex manifolds. Building on his joint work with Phong and Picard, the investigator will develop new analytic techniques for further understanding of this type of equation and to provide a complete answer to the question raised by Fu and Yau. Another major goal of this project is to study the anomaly flow, which was introduced by the investigator and his coauthors, aiming to develop analytical methods for solving the Hull-Strominger system on general three dimensional non-Kahler Calabi-Yau manifolds. Short-time existence of solutions of the flow were obtained in the joint work with Phong and Picard. The investigator will study the long-time behavior and convergence of this flow, and will also use this flow investigate the relation between the balanced cone and the Kahler cone on Kahler manifolds. To accomplish these goals, the investigator will develop new tools for the study of nonlinear elliptic and parabolic partial differential equations without concavity property.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.