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Nonlinear PDEs and complex geometry

非线性偏微分方程和复杂几何

基本信息

批准号：
1406164
负责人：
Benjamin Weinkove
金额：
$ 18.53万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406164&HistoricalAwards=false
关键词：
Nonlinear PDEs complex geometry

项目摘要

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.

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组来描述的。几何分析家使用非线性微分方程式试图对几何空间中可能出现的结构进行分类和理解。一个被称为Ricci流的方程已经被成功地用来证明每个三维空间都可以被分割成碎片，每个碎片都有一个众所周知的几何图形。Ricci流的工作原理是通过非线性热流过程演变出一个度量，该度量定义了两点之间的距离的概念。然而，对于四维或更高维度的空间(如我们生活的时空)，这个过程还没有被很好地理解。这个项目的一个主要目标是研究四维和更高维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。这个项目关注的是被赋予了“复杂结构”的空间。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出现的奇点的种类。复杂的几何图形出现在物理理论中，例如弦理论。这个项目将使用一种名为Chern-Ricci流的新热流方程，在四个维度上研究复杂的几何形状。在高维空间中，一个类似于弦理论中研究较多的Calabi-Yau方程的方程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组，重点是非Kahler空间。PI将建立在复杂表面上Kahler-Ricci流的结果基础上，以了解Chern-Ricci流的行为，即使在非Kahler的情况下，PDE也是有意义的。这个项目将研究与复杂表面分类有关的Chern-Ricci流的行为。特别是，PI将研究这种流动如何收缩特殊的曲线，以及这种流动在VII类曲面上的行为。对于高维非Kahler复流形，PI将使用复Monge-Ampere方程来研究上同调和Kahler流的存在性问题。此外，PI还将研究一个新的关于(n-1)重亚调和函数的Monge-Ampere方程，该方程与复流形上的Gauduchon度量和平衡度量有关。这个新方程的解将解决一个由来已久的Gauduchon猜想，并可能应用于射影簇的变形问题。最后，PI打算将这些思想推广到具有相容的几乎复杂结构的辛四维流形上的Donaldson的Monge-Ampere型方程。Donaldson关于该方程解的存在性的一个猜想可以归结为二阶估计。如果这个估计成立，它将应用于4-流形的辛拓扑。