Nonlinear Partial Differential Equations and Geometry

许多物理理论已经成功地被称为偏微分方程(PDE)的数学方程所模拟。经典的例子是热方程、波动方程和拉普拉斯方程，它们分别描述了理想条件下热、波和静电势的行为。这些是线性偏微分方程组。更复杂的理论往往用非线性偏微分方程组来建模。经典的非线性偏微分方程组是Monge-Ampere方程，它与光学的物理理论有关，但也是研究几何的有力工具。本研究将研究几种与Monge-Ampere方程有关的非线性偏微分方程组，这些偏微分方程具几何性质或具有几何行为。研究的目标有两个：一是利用非线性偏微分方程组促进我们对几何空间及其结构的理解；二是利用几何的工具和语言来理解非线性偏微分方程组的解的现象和行为。此外，PI还将对研究生进行非线性偏微分方程组和几何方法的培训，并指导他们的研究。有五个项目，通过复杂的Monge-Ampere方程的共同主题联系在一起，重点是开发新的分析工具和战略，以得出先验估计。对于第一个项目，PI将研究Donaldson的猜想，将复Monge-Ampere方程上的Yau定理推广到辛四维流形上，使用ANSATZ将非线性偏微分方程组简化为单个实值函数的方程。第二个项目将研究佩雷尔曼对Kahler-Ricci流的估计，目的是了解存在有限时间奇点时流的行为。另一个项目是研究这种流的非Kahler版本，称为Chern-Ricci流，重点是复杂表面上的有限时间奇点。另一个项目将扩展Chen-cheng和Shenin在非Kahler复杂曲面上关于具有常陈标曲率的度量的存在性问题的工作。最后一个项目是研究满足某些结构条件的偏微分方程凸解的一致凸性，建立在不变秩定理的基础上。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many physical theories have been modeled successfully by mathematical equations known as partial differential equations (PDEs). The classical examples are the heat equation, the wave equation and the Laplace equation, which describe in ideal conditions the behavior of heat, waves and electrostatic potentials respectively. These are linear PDEs. More complicated theories are often modeled by nonlinear PDEs. A classic nonlinear PDE is the Monge-Ampere equation which has connections to the physical theory of optics but is also a powerful tool in the study of geometry. This research will investigate several nonlinear PDE, all related to the Monge-Ampere equation, which arise in geometry or which exhibit geometric behavior. The research goals are two-fold: to use nonlinear PDEs to advance our understanding of geometric spaces and the structures that live on them; and to understand the phenomena and behavior of solutions of nonlinear PDE using the tools and language of geometry. In addition the PI will also train graduate students in the methods of nonlinear PDEs and geometry, and guide their research.The PI will carry out research on nonlinear PDEs and geometry. There are five projects, linked by the common theme of the complex Monge-Ampere equation and focusing on developing new analytic tools and strategies to derive a priori estimates. For the first project, the PI will investigate a conjecture of Donaldson extending Yau’s theorem on the complex Monge-Ampere equation to the setting of symplectic 4-manifolds, using an ansatz which reduces the nonlinear PDE to an equation of a single real-valued function. A second project will study Perelman’s estimates for the Kahler-Ricci flow with the goal of understanding the behavior of the flow when there is a finite time singularity. A further project is to investigate a non-Kahler version of this flow, known as the Chern-Ricci flow, with a focus on finite time singularities on complex surfaces. Another project will extend the work of Chen-Cheng and Shen on the question of existence of metrics with constant Chern scalar curvature, in the setting of non-Kahler complex surfaces. A final project is to study the uniform convexity of convex solutions to PDEs satisfying certain structure conditions, building on constant rank theorems.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.