Variational Problems and Nonlinear Equations in Geometry

几何中的变分问题和非线性方程

• 批准号: 1509633
• 负责人: Matthew Gursky
• 金额: $ 19.73万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1509633&HistoricalAwards=false
• 关键词: Variational; Problems; Nonlinear; Equations; Geometry

The interaction of geometry and analysis dates back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world such as angles, distances and properties of certain shapes. Differential geometry was developed to use the tools of calculus to understand the geometry of curved spaces--for example, the curvature of space by matter as predicted by general relativity, or the properties of soap bubbles (which turn out to be related to the equations describing black holes). In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study. The functional determinant of an elliptic operator is a problem originating in spectral theory and mathematical physics, and the analysis of the particular problem the PI considers is a variational problem leading to a fourth order elliptic equation. The associated Lagrangian is unbounded, and the existence of solutions and their qualitative properties is highly nontrivial. Similar equations are used to model the properties of thin films. Another problem the PI studies concerns the moduli space of Riemannian metrics that are critical points of functionals given by the integral of quadratic curvature quantities. In work of the PI with J. Viaclovsky, they constructed new examples of critical points, but this construction naturally leads to various conjectures about the moduli space of solutions. An important example of a quadratic curvature functional is the Weyl functional. Critical points are called Bach-flat metrics, and include important examples such as self-dual metrics. One moduli space problem the PI studies is a question of Singer about the linearized problem for self-dual manifolds of positive scalar curvature.

几何和分析的相互作用至少可以追溯到18世纪，但仍然是数学研究的重要且高度活跃的领域。几何学的经典主题源于我们希望理解物理世界的某些特性的愿望，例如某些形状的角度，距离和特性。开发了差异几何形状来使用微积分的工具来了解弯曲的空间的几何形状 - 例如，按一般相对性预测的空间曲率或肥皂气泡的性质（结果与描述黑洞的方程相关）。就像笛卡尔意识到可以使用代数研究平面几何形状一样，可以使用分析中的技术（尤其是微分方程）研究差异几何形状。 该项目的研究涉及几何和数学物理学的不同问题，但与数学分析在他们的研究中所扮演的角色结合在一起。 椭圆运算符的功能决定因素是源于光谱理论和数学物理学的问题，对特定问题的分析PI认为是一个变异问题，导致了第四阶椭圆方程。相关的Lagrangian是无限的，解决方案的存在及其定性特性是高度不平凡的。类似的方程式用于对薄膜的性质进行建模。 PI研究的另一个问题涉及Riemannian指标的模量空间，这是二次弯曲量积分给出的功能的关键点。在PI与J. Viaclovsky的工作中，他们构建了关键点的新例子，但是这种结构自然会导致有关解决方案模量空间的各种猜想。二次曲率功能的一个重要例子是Weyl功能。临界点称为Bach-Flat指标，包括重要示例，例如自偶像指标。一个模量空间问题PI研究是关于正与标量曲率的自与双流形的线性化问题的唱片的问题。