Nonlinear PDEs in Complex Geometry and Physics

复杂几何和物理中的非线性偏微分方程

• 批准号: RGPIN-2021-02600
• 负责人: Picard, Sebastien
• 金额: $ 1.89万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758559
• 关键词: Nonlinear; PDEs; Complex; Geometry; Physics

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.

这项研究的目的是进一步发展我们的知识微分几何和非线性偏微分方程。我们的方法是使用非线性偏微分方程的分析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一个关于其曲率张量的约束方程，该方程可以表示为一个非线性偏微分方程。我特别感兴趣的方程来自数学物理和复杂的几何，在卡勒和非卡勒设置。黎曼度量可以用来描述底层空间的原理在数学中随处可见，从复分析的单值化定理开始，此后出现在稳定向量丛上的厄米-杨-米尔斯联络理论、科代拉嵌入定理、庞加莱猜想和姚-田-唐纳森猜想中，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。除了物理应用之外，曲率约束下的度量研究还将几何学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，椭圆和抛物微分方程新技术的发展往往导致微分几何的突破。第一个项目是为Hull-Strominger系统建造新的解决方案。这是理论物理学家提出的一个微分方程组，作为杂化弦的模型;此外，所涉及的厄米度量可能具有非零挠率，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。这种几何流可以被视为适应于有挠卡拉比-丘流形几何设置的里奇流的模拟。第三个项目涉及的纯偏微分方程的问题，获得一个先验估计的某些完全非线性椭圆方程的解决方案。这些方程的灵感来自于几何学，例子包括复杂的Monge-Ampere方程和第k个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，主要关注理论物理方程。