Scalar curvature and geometric variational problems

标量曲率和几何变分问题

• 批准号: 2303624
• 负责人: Chao Li
• 金额: $ 38.05万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303624&HistoricalAwards=false
• 关键词: Scalar; curvature; geometric; variational; problems

Curvature describes local bending of a space, and is used to distinguish how two shapes are different. This project concerns a particular notion of curvature, called the scalar curvature. Scalar curvature determines 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. Curvature arises throughout the natural sciences, and particularly in general relativity, where scalar curvature is the Lagrangian density of the Einstein-Hilbert action. A natural yet deep question is to understand the effects of scalar curvature on the global properties of a manifold. In particular, the project will focus on geometric variational problems, including minimal surfaces and soap bubbles. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. Some key questions include the existence, regularity and topology of minimal surfaces. These two aspects of the research are deeply related, and will advance our understanding of the shape of nature. The PI will integrate this research with a variety of knowledge-disseminating activities that include organizing seminars, conferences, mini-courses, and reading groups, and giving public lectures. The proposed research concerns a range of topics in differential geometry, geometric measure theory and partial differential equations. Particularly, the PI would like to focus on the following four main topics. The first topic is the investigation of the obstruction problem for manifolds with positive scalar curvature, including the well-known `K(pi, 1) conjecture’. The second topic is to further understand the geometric comparison theorem for scalar curvature lower bound using Riemannian polyhedra. Minimal surface and soap bubbles are key technical tools for such problems. The third topic is to further investigate the stable Bernstein problem for minimal surfaces in R^n. The fourth topic is to further study the moduli space of positive scalar curvature metrics on 3-manifolds (with or without boundary). Understanding such questions has potential applications in 4-dimensional general relativity and in topology.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.

曲率描述了一个空间的局部弯曲，并用于区分两个形状的不同。该项目涉及一个特定的曲率概念，称为标量曲率。标态曲率确定了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。曲率在整个自然科学，尤其是在一般相对论中都产生，其中标态曲率是爱因斯坦 - 希尔伯特作用的拉格朗日密度。一个自然而深的问题是了解标量曲率对流形全球性质的影响。特别是，该项目将集中于几何变异问题，包括最小表面和肥皂气泡。最小的表面是自然界许多接口的数学模型。总体而言，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。一些关键问题包括最小表面的存在，规律性和拓扑。这项研究的这两个方面是密切相关的，并将促进我们对自然形状的理解。 PI将将这项研究与各种知识截止性活动相结合，包括组织半手，会议，迷你场和阅读小组以及进行公开讲座。拟议的研究涉及差异几何，几何测量理论和部分微分方程的一系列主题。部分地，PI希望专注于以下四个主要主题。第一个主题是对具有积极标态曲率的流形的客观问题的调查，包括众所周知的“ k（pi，1）概念”。第二个主题是进一步理解使用里曼polyhedra的标态曲率下限的几何比较理论。最小的表面和肥皂气泡是解决此类问题的关键技术工具。第三个主题是进一步研究r^n中最小表面的稳定伯恩斯坦问题。第四个主题是进一步研究3个manifolds（有或没有边界）上正标度曲率指标的模量空间。理解此类问题在4维常规可靠性和拓扑中具有潜在的应用。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。