Nonlinear PDEs in Complex Geometry and Physics

本研究旨在进一步发展我们在微分几何和非线性偏微分方程方面的知识。我们的方法是利用非线性偏微分方程分析的技术来研究流形上的黎曼度量张量。最优黎曼度量通常满足曲率张量的约束方程，该约束方程可以表示为非线性偏微分方程。我对来自数学物理和复杂几何的方程特别感兴趣，无论是在Kahler还是非Kahler环境下。黎曼度量可以用来描述底层空间的原理在整个数学中都可以找到，从复分析的均匀化定理开始，到稳定向量束上的Hermitian-Yang-Mills连接理论，再到Kodaira嵌入定理，庞加莱猜想和you - tian- donaldson猜想，仅举几例。这一提议延续了这一传统，同时引入了理论物理中引入的新方程。在物理应用之外，曲率约束下度量的研究将几何和偏微分方程领域联系起来。一方面，微分几何提供了具有深层结构性质的非线性方程的有趣例子，另一方面，椭圆型和抛物型微分方程新技术的发展往往导致微分几何的突破。第一个项目涉及赫尔-施特罗明格系统的新解决方案的构建。这是一个由理论物理学家提出的微分方程组，作为异质弦的模型；此外，所涉及的厄米度量可能具有非零扭转，这使得它们从非卡勒复几何的角度来看很有趣。第二个项目涉及异常流的分析。这种几何流可以看作是里奇流的一种模拟，适合于具有扭转的Calabi-Yau流形的几何设置。第三个项目涉及纯偏微分方程问题，即获得某些完全非线性椭圆方程解的先验估计。这些方程的灵感来自几何，例子包括复杂的蒙日-安培方程和k- Hessian方程。总之，本研究的目标是推进微分几何中非线性方程的分析，主要集中在理论物理方程上。

This proposed research aims to further develop our knowledge of differential geometry and nonlinear partial differential equations. Our approach is to use techniques from the analysis of nonlinear partial differential equations to study Riemannian metric tensors on manifolds. Optimal Riemannian metrics generally satisfy a constraint equation on their curvature tensor which can be expressed as a nonlinear PDE. I am particularly interested in equations coming from mathematical physics and complex geometry, in both Kahler and non-Kahler settings. The principle that Riemannian metrics can be used to describe the underlying space is found throughout mathematics, beginning with the uniformization theorem of complex analysis, and since appearing in the theory of Hermitian-Yang-Mills connections on stable vector bundles, the Kodaira embedding theorem, the Poincaré conjecture, and the Yau-Tian-Donaldson conjecture, just to name a few. This proposal continues this tradition while bringing in new equations introduced in theoretical physics. Beyond physical applications, the study of metrics subject to a curvature constraint links geometry and the field of partial differential equations. On one hand, differential geometry provides interesting examples of nonlinear equations with deep structural properties, and on the other hand, the development of new techniques in elliptic and parabolic differential equations often leads to breakthroughs in differential geometry. The first project concerns the construction of new solutions to the Hull-Strominger system. This is a system of differential equations proposed by theoretical physicists as a model for the heterotic string; furthermore, the Hermitian metrics involved may have nonzero torsion, which makes them interesting from the point of view of non-Kahler complex geometry. The second project concerns the analysis of the Anomaly flow. This geometric flow can be viewed as an analog of the Ricci flow adapted to the geometric setting of Calabi-Yau manifolds with torsion. The third project concerns the pure PDE problem of obtaining a priori estimates on solutions of certain fully nonlinear elliptic equations. These equations are inspired by geometry, and examples include the complex Monge-Ampere equation and the k-th Hessian equation. In summary, the goal of this research is to advance the analysis of nonlinear equations in differential geometry, with a main focus on equations from theoretical physics.