Geometric Variational Problems and Scalar Curvature

几何变分问题和标量曲率

• 批准号: 2005287
• 负责人: Chao Li
• 金额: $ 17.45万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005287&HistoricalAwards=false
• 关键词: Geometric; Variational; Problems; Scalar; Curvature

One aspect of the proposed research has to do with the geometry and topology of manifolds with scalar curvature lower bounds. Scalar curvature is the simplest curvature invariant of a Riemannian manifold. It represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. Scalar curvature also arises in natural sciences. For instance, in general relativity, it is the Lagrangian density of the Einstein-Hilbert action. A natural and deep question in geometry, topology and mathematical physics is to understand the affect of scalar curvature conditions on a manifold. The other main area of investigation concerns minimal surfaces. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In mathematical model of general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. The PI will investigate the existence, regularity and topology of minimal surfaces. The two aspects proposed here are deeply connected via geometric variational theory.The project concerns a range topics on differential geometry, geometric measure theory and partial differential equations. A main theme of the research in geometry will be a geometric comparison theorem for scalar curvature using Riemannian polyhedra, with the aim to define weak notions of positive scalar curvature on spaces with low regularity. The PI plans to continue his investigations into such a theorem for more general polytopes, especially simplexes of higher dimensions, and its connection to quasi-local mass in general relativity. The PI also plans to continue his investigation on the structure of moduli spaces of manifolds with positive scalar curvature and mean convex boundary, including studying its high homotopy groups, and the structure of moduli spaces defined by other related curvature conditions. In addition, the PI will study singular spaces with scalar curvature lower bounds, and understand when such a singular manifold arises as a certain limit of smooth manifolds with same assumptions. A central tool in the PI’s research is the theory of minimal varieties. The PI plans to understand the existence and regularity of minimal surfaces with free boundary and capillary boundary conditions in general Lipschitz domains, especially in locally convex polyhedral domains. He also plans to establish a general existence theory of capillary surfaces via a min-max construction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

拟议的研究的一个方面与标态曲率下限的流形的几何形状和拓扑有关。标量曲率是riemannian歧管的最简单曲率不变。它代表了riemannian歧管中小的大地球的体积与欧几里得空间中标准球的体积偏离。自然科学也出现了标态曲率。例如，从一般相对论中，这是爱因斯坦 - 希尔伯特作用的拉格朗日密度。几何，拓扑和数学物理学中的自然和深度问题是了解标量曲率条件对多种状态的影响。投资的另一个主要领域涉及最小的表面。最小的表面是自然界许多接口的数学模型。在一般相对性的数学模型中，最小表面是黑洞的“明显范围”。肥皂膜和毛细管界面还提供了最小表面的示例。 PI将研究最小表面的存在，规律性和拓扑。这里提出的两个方面通过几何变异理论深入联系。该项目涉及差异几何，几何测量理论和部分微分方程的范围主题。几何研究的主要主题是使用Riemannian Polyhedra的标量曲率的几何比较理论，目的是在较低规律性的空间上定义弱标度曲率的弱音。 PI计划继续投资于此类定理，以获取更通用的多面有，尤其是更高维度的单纯性，以及它与一般可靠性的准本地质量的联系。 PI还计划继续对具有积极标态曲率和平均凸边界的多种歧管空间的结构进行投资，包括研究其高同型组，以及由其他相关曲率条件定义的模量空间的结构。此外，PI将研究具有标态曲率下边界的奇异空间，并了解何时出现这种奇异的歧管是具有相同假设的平滑歧管的一定极限。 PI研究中的一个核心工具是最小变化的理论。 PI计划了解一般Lipschitz域中具有自由边界和毛细管边界条件的最小表面的存在和规律性，尤其是在局部凸的多面体结构域中。他还计划通过Min-Max的建设建立毛细血管表面的一般存在理论。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响标准，被视为通过评估而被视为珍贵的支持。