Nonlinear PDEs and complex geometry

许多物理定律，如爱因斯坦场方程，都是用几何空间上的非线性微分方程来描述的。 几何分析师使用非线性微分方程来尝试分类和理解几何空间中可能出现的结构。 一个被称为Ricci流的方程已经被成功地用来表明，每个三维空间都可以被切割成碎片，每个碎片都有一个很好理解的几何形状。 里奇流的工作原理是通过一个非线性热流过程来演化一个度量，该度量定义了两点之间的距离。 然而，对于四维或更高维的空间（例如我们生活的时空），这个过程还没有得到很好的理解。 该项目的一个主要目标是研究四维及更高维度空间上的非线性方程，目的是对这些空间进行分类，并了解生活在这些空间上的自然几何结构。 这个项目的重点是赋予空间“复杂的结构”。 尽管有这个名字，但具有复杂结构的空间更容易研究，因为这种额外的结构限制了可能发生的奇异性。 复杂的几何出现在物理理论中，例如弦理论。 该项目将使用称为Chern-Ricci流的新热流方程研究四维复杂几何。 在更高的维度，一个类似于卡-丘方程的方程（在弦理论中被广泛研究）将被用来研究这些空间的结构。这个项目将研究复杂几何中的非线性偏微分方程，重点是非Kahler空间。 PI将建立在复杂表面上的Kahler-Ricci流的结果上，以理解Chern-Ricci流的行为，这是一种即使在非Kahler情况下也有意义的PDE。 这个项目将研究与复杂表面分类相关的陈-里奇流的行为。 特别是，PI将研究这种流动如何收缩特殊曲线，以及流动在VII类表面上的行为。 对于高维非Kahler复流形，PI将使用复Monge-Ampere方程来研究Kahler流的上同调和存在性问题。 此外，PI将研究一个新的Monge-Ampere方程的（n-1）-plurisubharmonic函数，这是相关的Gauduchon和平衡度量的复流形上。 这个新方程的解决方案将解决一个长期存在的猜想Gauduchon和有可能的应用变形问题的投影品种。 最后，PI打算将这些想法扩展到具有相容几乎复结构的辛4-流形上的Monge-Ampere型唐纳森方程。 唐纳森关于该方程解的存在性的一个猜想可以归结为一个二阶估计。 如果这个估计成立，它将应用于4-流形的辛拓扑。

Many of the laws of physics, such as the Einstein field equations, are described using nonlinear differential equations on geometric spaces. Geometric analysts use nonlinear differential equations to try to classify and understand the possible structures that can occur in geometric spaces. An equation known as the Ricci flow has been used to great success to show that every three dimensional space can be cut up into pieces, each of which has a well-understood geometry. The Ricci flow works by evolving a metric, which defines a notion of distance between two points, by a nonlinear heat flow process. However, for spaces of dimension four or higher (such as the space-time we live in), this process is not yet well understood. A major goal of this project is to investigate nonlinear equations on spaces of dimension four and higher with the aim of classifying these spaces and understanding the natural geometric structures that live on them. This project focuses on spaces endowed with a 'complex structure'. Despite the name, spaces with complex structures are easier to study, since this additional structure limits the kinds of singularities that could occur. Complex geometries appear in physical theories, such as string theory. This project will investigate complex geometries in four dimensions, using a new heat flow equation known as the Chern-Ricci flow. In higher dimensions, an equation analogous to the Calabi-Yau equation (much studied in string theory) will be used to investigate the structure of these spaces.This project will investigate nonlinear PDEs in complex geometry, with a focus on spaces which are non-Kahler. The PI will build on results for the Kahler-Ricci flow on complex surfaces to understand the behavior of the Chern-Ricci flow, a PDE which makes sense even in the non-Kahler case. This project will investigate the behavior of the Chern-Ricci flow in relation to the classification of complex surfaces. In particular, the PI will study how exceptional curves are contracted by this flow, and how the flow behaves on Class VII surfaces. For higher dimensional non-Kahler complex manifolds, the PI will use the complex Monge-Ampere equation to study questions about cohomology and existence of Kahler currents. In addition, the PI will study a new Monge-Ampere equation for (n-1)-plurisubharmonic functions, which is related to Gauduchon and balanced metrics on complex manifolds. A solution to this new equation will solve a long-standing conjecture of Gauduchon and have possible applications to deformation problems for projective varieties. Finally, the PI intends to extend these ideas to a Monge-Ampere type equation of Donaldson on symplectic 4-manifolds with compatible almost complex structures. A conjecture of Donaldson on existence of solutions to this equation can be reduced to a second order estimate. If this estimate holds it would give applications to the symplectic topology of 4-manifolds.