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.

所提出的研究的一个方面与具有标量曲率下界的流形的几何和拓扑有关。标量曲率是黎曼流形中最简单的曲率不变量。它表示黎曼流形中一个小测地线球的体积与欧几里德空间中标准球的体积的偏离量。标量曲率也出现在自然科学中。例如，在广义相对论中，它是爱因斯坦-希尔伯特作用的拉格朗日密度。理解标量曲率条件对流形的影响是几何、拓扑和数学物理中一个自然而深刻的问题。另一个主要研究领域涉及最小表面。最小曲面是自然界中许多界面的数学模型。在广义相对论的数学模型中，最小表面是黑洞的“视视界”；肥皂膜和毛细管界面也提供了最小表面的例子。PI将研究最小曲面的存在性、规律性和拓扑结构。这两个方面是通过几何变分理论紧密联系在一起的。该项目涉及微分几何、几何测度理论和偏微分方程等一系列主题。几何研究的一个主要主题是使用黎曼多面体的标量曲率的几何比较定理，目的是定义低正则性空间上正标量曲率的弱概念。PI计划继续研究更一般的多面体，特别是高维的简单体，以及它与广义相对论中准局部质量的联系。PI还计划继续研究具有正标量曲率和平均凸边界的流形的模空间结构，包括研究其高同伦群，以及由其他相关曲率条件定义的模空间结构。此外，PI将研究具有标量曲率下界的奇异空间，并了解在相同假设下，这种奇异流形何时作为光滑流形的某一极限出现。PI研究的一个核心工具是最小变量理论。PI计划了解一般Lipschitz域中，特别是局部凸多面体域中具有自由边界和毛细边界条件的最小曲面的存在性和规律性。他还计划通过最小-最大构造建立毛细表面的一般存在理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。